Year 12 Maths A Textbook - Chapter 4

4syllabussyllabusrrefefererenceenceStrand:Statistics and probability

Core topic:Exploring and understanding data

In thisIn this chachapterpter4A Populations and samples4B Samples and sampling4C Bias4D Contingency tables4E Applications of statistics and

probability

Populations, samples, statistics and probability

Maths A Yr 12 - Ch. 04 Page 167 Wednesday, September 11, 2002 4:07 PM

168

M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d

Populations and samples

Early population counts were musters, where community members were gatheredand counted. In 1828, the first Australian

census

was conducted in New SouthWales. Each State conducted its own separate census until 1886, five years afterthe first simultaneous census of the British Empire. In 1901, a common censuswas conducted throughout Australia; however, the results were not collated toform a total for Australia.

The Census and Statistics Act of December 1905 provided that: ‘

TheCensus shall be taken in the year 1911, and in every tenth year thereafter

.’During the Depression and World War II, no census was taken. The first

post-war census took place in Australia in 1947.The types of questions have changed over time to reflect the changes in our

society. The time required to process the responses to the questions has beenreduced with the introduction of Optical Mark Reading machines (1991 census)

and Intelligent Character Recognition machines which can read handwritten wordsin the 2001 census. Since 1961, a census has been held every five years, and the

fourteenth national Census of Housing and Population was held on 7 August 2001.The 2001 census coincided with Australia’s Centenary of Federation. Participants

were given the opportunity to place their census forms in a time capsule (to be held bythe National Archives) for 99 years. Descendants would then have a glimpse into thelives of their forebears.

Many of the skills required for this chapter were developed in Year 11 (chapters 9 and 10of

Maths Quest Maths A Year 11

). Revise the methods by completing the following exercises.

1

Write each of the following as a decimal (correct to 3 decimal places).

a b c d

2

Convert each of the following to percentages.

a b

0.125

c d

0.04

3

Use your calculator to generate a set of 10 random integers in the range:

a

1 to 20 inclusive

b

50 to 100 inclusive.

4

Round the following numbers to integers.

a

3.6

b

4.02

c

2.91

d

6.5

e

0.9

5

Find the unknown in each of the following.

a

=

b

=

c

=

d

=

e

=

6

What types of features on a graph can cause it to be misleading?

7

For the following sets of scores

x

:6, 9, 8, 7, 6, 5, 8, 11, 6, 7

Calculate:

a

Σ

x

b

x–

c

median

d

mode

e

lower quartile

f

upper quartile

g

range

h

inter-quartile range

.

38--- 1

12------ 65

80------ 124

210---------

34--- 85

200---------

14--- 2

a--- 3

7--- b

21------ 2

9--- 5

c--- 5

d--- 2

7--- f

9--- 7

6---

WorkS

HEET 4.1

Maths A Yr 12 - Ch. 04 Page 168 Friday, September 13, 2002 9:19 AM

C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y

169

Populations

A census represents information or data collected from every member of the popu-lation. The term

population

does not necessarily represent a group of people; it is alsoused to represent a group of objects with the same defined characteristics. So, the popu-lation under study may be the wildlife in a national forest, the number of wattle trees ina park, the soil in a farmer’s field or the number of cars in a country town.

In some cases it may be possible to determine the exact extent of the population (thenumber of wattle trees in the park or the number of cars in a country town); however, itis often not possible to obtain an exact figure for the population (the extent of the wild-life in a forest) because circumstances are constantly changing.

Sometimes it is not physically possible to consider the whole population (all the soilin a farmer’s field), as it would not be practical. It is often very costly and time con-suming to consider the whole population in a study. For these reasons, we need toobtain information about the population by selecting a

sample

that can then be studied.A

census

is conducted when we obtain information from the whole

population

;however, a

survey

is conducted on a

sample

of the population.

Australia’s population and housing census

It is important that we understand the reason for recording statistical data accurately. In our society, it is difficult to imagine a world without statistics. Try to imagine a State of Origin football match where no one kept the score! The excitement of the game would probably hold our attention for a while, but if no score was recorded, winning or losing would not be an issue, and we would soon lose interest.

A census is an example of information collected from the whole

population

. It is not always possible or feasible to conduct a

questionnaire

on the whole population, so when this opportunity arises, it is vital to ensure that the questions are carefully worded and that relevant information is sought.

The Australian Bureau of Statistics (ABS) is the government department responsible for administering the Australian census, then collating and analysing the responses. Their website <www.jaconline.com.au/maths/weblinks> details information about their role and it displays statistical data from many areas. Access this site to conduct your research. Prepare a report providing responses to the following:

1

What is a national housing and population census?

2

Who takes part?

3

Is it compulsory to take part?

4

What types of questions are asked in the census? How have they changed over the years?

5

Why should we have a census?

6

Who has access to the information we provide?

7

How is the census conducted?

8

Conclude your report with an expression of your opinion (agreement/disagreement) of the answers gathered from your research. Provide constructive suggestions to improve any aspect of the gathering, collating and analysing of the census data.

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stigationinvestigatio

n


http://www.jaconline.com.au/maths/weblinks

170


The particular circumstances determine the status of thebody being studied (whether it represents the population or asample of the population). Consider, for example, your Math-ematics A class. If we were to try to determine the number ofleft-handed people in your school who studied MathematicsA, and there was only one such class in your school, thenyour class would be regarded as the whole population. If, onthe other hand, there were several Mathematics A classes inyour school, then your class would be considered a sample ofthe population.

Samples

It is most important when selecting a sample from a popu-lation that the sample represents the population as closely aspossible. For this to occur, the characteristics of the sampleshould occur in the same proportions as they do in the popu-lation. There is little point in selecting a sample where this isnot the case, for analysis of the sample would lead to mis-leading conclusions. We often see this occurring when pollsare conducted prior to an election. Quite frequently they pre-dict a particular outcome while the election results in a dif-ferent outcome.

In each of the following, state if the information was obtained by census or survey.a A school uses the roll to count the number of students absent each day.b The television ratings, in which 2000 families complete a questionnaire on what they

watch over a one-week period.c A light globe manufacturer tests every hundredth light globe off the production line.d A teacher records the examination marks of her class.

THINK WRITE

a Every student is counted at roll call each morning.

a Census

b Not every family is asked to complete a ratings questionnaire.

b Survey

c Not every light globe is tested. c Surveyd The marks of every student are recorded. d Census

1WORKEDExample

remember1. Before beginning a statistical investigation it is important to identify the target

population.2. The information can be obtained either by:

(a) Census — the entire target population is questioned, or(b) Survey — a population sample is questioned such that those selected are

representative of the entire target population.

remember


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 171

Populations and samples

1 Copy and complete the following:When we obtain data from the whole population, we conduct a _______________;however, a survey obtains data from a _______________ of the population.

2 A school conducts an election for a new school captain.Every teacher and student in the school votes. Is this anexample of a census or a survey? Explain your answer.

3 A questionnaire is conducted by a council to see what sportingfacilities the community needs. If 500 people who live in thecommunity are surveyed, is this an example of a census or asurvey?

4 For each of the following, state whether a census or a surveyhas been used.a Two hundred people in a shopping centre are asked to

nominate the supermarket where they do most of their gro-cery shopping.

b To find the most popular new car on the road, 500 new carbuyers are asked what make and model car they purchased.

c To find the most popular new car on the road, the make andmodel of every new car registered are recorded.

d To find the average mark in the mathematics half-yearlyexamination, every student’s mark is recorded.

e To test the quality of tyres on a production line, every 100thtyre is road tested.

5 For each of the following, recommend whether you would usea census or a survey to find:

a the most popular television program on Monday night at 7.30 pmb the number of cars sold during a period of one yearc the number of cars that pass through the tollgates on the Brisbane Gateway Bridge

each dayd the percentage of defective computers produced by a company.

6 An opinion poll is conducted to try to predict the outcome of an election. Two thou-sand people are telephoned and asked about their voting intention. Is this an exampleof a census or a survey?

Samples and samplingWhen we select a sample from a population, if it has been chosen carefully, it should,upon analysis of the data, yield the same (or very similar) results to those of the population.A decision must be made regarding the size of the sample. In practice, the size chosen isthe smallest one that would be considered appropriate in those circumstances and the sizethat would yield a proportion of the elements close to that occurring in the population.

4AWORKEDExample

1


172


Sampling methods

Several techniques can be employed to select a sample from a population. Somecommon methods are

random sampling

,

accessibility sampling

,

systematicsampling

,

quota sampling

,

judgmental sampling

,

stratified sampling

,

clustersampling

, and

capture–recapture sampling

.

Sample size

The aim of this investigation is to observe how the composition of a sample is affected by the sample size.

1

Take a large packet of mixed coloured jellybeans (200 or more) as the population. (Coloured disks could be substituted.)

2

Place the jellybeans in a container and mix well. Without looking, draw out a sample of 10 in such a way that each jellybean has an equal chance of being selected. This can then be considered a random sample. Count the number of red jellybeans in the sample of 10.

3

Return the sample of 10 jellybeans to the container, mixing them well with the others. Select a random sample of 20 jellybeans, using the same method as before; record the number of red ones.

4

Continue in this manner, returning each sample to the container, mixing them well, then selecting a sample containing 10 more than the previous selection. Record the number of red jellybeans in each of the samples.

5

Generate a table of the format below:

6

Enter the data in the first and third columns (sample size and proportion) into a spreadsheet or graphics calculator. Graph the sample size against the proportion of red jellybeans. (Alternatively, this could be graphed on graph paper.)

7

Knowing that the proportion of red jellybeans in the whole population (the final row in the table above) represents the true answer, comment on the effect of the sample size on the composition of the sample.

8

For your particular experiment, what would be the minimum sample size which closely resembles the composition of the population?

9

If a sample is used to predict the composition or characteristics of a population, describe what you feel are the desirable qualities of the sample in order to be a reliable predictor of the composition or characteristics of the population.

10

Repeat the experiment. Comment on the similarities/differences in your results.

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Sample size Number of red jellybeans Proportion of red jellybeans

10 (as a decimal)

20

30

…200

Whole population

Maths A Yr 12 - Ch. 04 Page 172 Friday, September 13, 2002 9:19 AM


Random samplingA simple random sample is one for which each element of the population has an equalchance of being chosen. A way in which this can be achieved is by numbering eachelement of the population then randomly selecting items for the sample by usingrandom digit tables, the random function on a calculator or numbers drawn from acontainer.

Random samplingThe aim of this investigation is to compare different random sampling techniques as methods of selecting a sample that is representative of the population. Consider selecting a random sample of ten (10) students from your mathematics A class. (Your class is the population in this investigation. You may adjust the sample size if you wish.)

1. Select a characteristic that is present in some of your class members such as brown eyes, fair hair, height above 175 cm and so on.

2. Calculate and record the proportion of the population in your class with this characteristic.

3. Have the students number off 1, 2, 3, … until all students have a number. This number for our purposes may be regarded as the population number.

Task 1 Using random digit tables to select a sample

Tables of randomly generated digits are published. Below are samples of sets of two-, three- and four-digit random number tables. These tables are generally much larger than the extracts shown. For our purposes, this size will be sufficient.

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Two-digit random number table

16 79 43 59 41 16 39 29 11 12

13 54 24 09 46 24 93 53 28 82

25 56 61 15 97 82 65 77 94 82

85 41 99 74 09 05 98 89 72 10

71 51 35 29 52 52 89 02 92 96

02 81 92 89 17 08 04 63 43 03

84 67 19 23 43 11 05 17 08 07

36 36 72 21 86 99 28 41 24 22

23 04 78 05 33 01 66 06 04 57

80 22 99 14 89 15 65 19 06 25


174 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d

Three-digit random number table

Four-digit random number table

The following rules apply to the use of random digit tables.

Step 1 Begin at any position in the table (this position being chosen randomly).

Step 2 Move in any direction (vertically, horizontally) along a column or row.

Step 3 Continue moving in this direction, recording the numbers as you go.

Step 4 If you use a three-digit or four-digit table, and you require only one or two-digit numbers in your selection, you may choose to use the digit/s on the left, on the right, on the edges and so on. (Make some random choice, and then be consistent.)

Step 5 If the same number is repeated, do not record the number a second time.

Step 6 Continue recording until the required total has been reached.

382 093 530 260 651 344 157 738 522 592

452 981 272 886 907 683 894 946 831 521

557 374 900 425 461 145 098 792 793 388

694 914 642 153 901 642 100 851 365 840

435 104 419 685 626 383 326 376 246 586

851 474 369 272 566 488 420 696 272 547

869 681 282 129 194 236 467 014 699 196

895 662 376 612 435 080 818 396 572 809

282 274 363 903 771 370 799 277 636 313

464 680 859 249 093 848 370 303 661 495

4070 8145 3435 0891 8504 6691 5329 3729 6800 6262

7368 6927 7980 6625 7301 0145 8729 8145 5299 3951

8859 8070 3664 1177 1821 3729 0064 3715 8166 2427

3065 6791 3344 9357 2928 3807 7301 9513 0058 4049

6776 6603 1700 5233 0925 5817 3709 3213 1282 8856

7977 8319 6074 7955 2059 4763 8885 8565 0755 5087

4843 0033 1948 2371 5640 9865 2105 5484 8890 6160

7678 3588 7213 5572 6939 2544 2461 3232 9394 0253

8521 9289 5756 9137 6540 5741 1777 2149 4079 5279

9895 0709 0323 7394 5003 2494 6829 4634 3586 6238



1 Use the two-digit random number table to select ten numbers within the range of numbers in your class.

2 Determine the students in your class to whom these numbers refer.

3 Calculate and record the proportion of these students with the characteristic you chose. How closely does it match the population proportion which you have previously calculated?

4 Repeat the experiment using the three-digit random number table, calculating and recording the proportion of students in this sample with your chosen characteristic.

5 Repeat using the four-digit random number table, again calculating and recording the proportion.

6 Compare the results obtained from your three samples with each other and with the population proportion. What conclusion/s can you form?

Task 2 Using the random function on a calculator1 Many scientific and graphing calculators can be set to generate random integers

in the range of your population number. (Your teacher will show you, if you are unsure.)

2 Use your calculator to generate ten different random integers.

3 Relate these numbers to specific students in your class.

4 Calculate and record the proportion of students in your sample with your chosen characteristic.

5 Compare this value with the population proportion.

Task 3 Using lot samplingThis type of sampling is used in drawing lotto winning numbers.

1 Write numbers (up to and including your population number) on small, equally sized pieces of paper and place them in a container. Mix well.

2 Draw the numbers one at a time (without replacing them) until ten numbers have been drawn.

3 Relate these numbers to the relevant students, as before.

4 Calculate and record the proportion of students with your chosen characteristic in this sample.

5 Compare the value with the population proportion.

Conclusions1 Draw up a table to display the results of all your experiments.

2 Compare the results obtained using the various techniques.

3 Did you find one method better than any other?

4 How did the results using these three methods compare with the population result?



Generating random integers using a spreadsheet

This activity creates a spreadsheet to generate random integers (whole numbers) within a given range. Consider the spreadsheet below.

1 Enter the headings in cells A1, A3, A4, A6, A7, A8 and A10.

2 Leave cells B7 and B8 blank. You will enter values in these cells once you run the spreadsheet.

3 In cell B11, enter the formula =INT(RAND()*($B$8-$B$7+1))+$B$7). This formula will generate a random integer in the range of the value entered in cell B7 to the value entered in cell B8 inclusive. (You will not find a correct value appears until you enter values in cells B7 and B8.)

4 Copy this formula to the region B11 to K20. This will generate 100 random integers in this region.

5 The function F9 will recalculate different sets of random integers. Add this instruction to cell A22.

6 Enter values in B7 and B8. Notice the set of integers produced. Press the F9 key to generate a different set. Continue to generate new sets, making sure that the numbers generated are within the range of those entered in cells B7 and B8. You will find that if you generate large integers you may have to widen columns B to K.

7 Save your spreadsheet and obtain a printout.

8 You may wish to use this spreadsheet for generating two-, three- and four-digit random number tables for your own use.

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Accessibility samplingThis method of sampling selects those items that are most accessible. Consider thefollowing:1. The student body in a school is investigating extended hours for the library. A

survey, conducted on students who were using the library after hours, overwhelm-ingly supported the proposal for extended hours.

2. The same survey, conducted on students in the gymnasium after school, indicated noneed for extending the library hours.

As can be seen from this example, bias can be introduced into the results of a survey bycarefully selecting the sample to either support or refute a cause. A sample drawn onlyfrom easily accessible items often does not represent the views of the population.

Systematic samplingIn this method, the sample items are selected using some system, such as every tenthitem, every item on the top left-hand corner of a page or every item in the fifth positionof a set of lists.

Consider a survey to determine the most popular service provider for Internetsubscribers. The sample could be selected by ringing every:• one hundredth name in the telephone directory• last name on each page• name at the top of each list of names on every page.Using the telephone directory to obtain survey data has the obvious disadvantage ofexcluding those who do not own a telephone and those with an unlisted telephonenumber. When using a systematic method as a sampling technique, best results areobtained if the population is first arranged randomly.

Use the three-digit random number table on page 174 to select ten students from a numbered class of 30 according to the following rules.Rule 1 Start in the bottom left-hand corner.Rule 2 Snake up and down the columns.Rule 3 Select the two digits on the right as the student number

Note: Use the three-digit random number table from page 174.

THINK WRITE

The selected numbers must be in the range 1 to 30 inclusive.Moving up the first column on the left, reading the last two digits, there are no numbers in the range. Continue by snaking down the second column and so on, until 10 numbers have been selected (ignore the second occurrence of a number).Give 10 student numbers. Students selected have numbers 14, 4, 19, 30,

25, 29, 12, 3, 26 and 1.

1

2

3

2WORKEDExample



Quota samplingThe quota technique specifies a particular number of items to be surveyed. Considerthe following scenario:

A group of businessmen is considering establishing a grammar school (no religiousaffiliations) in a town of approximately 50 000 people. They decide to conduct a marketsurvey on 1000 people. They specify the composition of the sample as follows.• The group should have 500 males and 500 females.• Within each of those groups, half should be school children and half adult.• The groups must contain people of all religious denominations — not in equal pro-

portions, as they do not occur equally in the community.• From the 500 children selected, there should be 200 from non-government schools

and 300 from government schools.Within these specified quotas, the person responsible for choosing the sample can

use any sampling strategy. This leaves the sample open to bias, depending on the integ-rity of those selecting the sample. It also enables substitutions to occur when thoseoriginally selected for the sample are not readily accessed.

Bearing in mind the problems associated with this type of sampling, this method canprove to be cost-effective and quite reliable in its predictions if the composition of thesample is appropriate and the sample is selected in an unbiased manner.

Judgmental samplingUsing this method of obtaining a sample, the person conducting the survey must makea judgment as to the composition of the sample. This obviously is reliant on the goodjudgment of those selecting the sample. Consider, for instance, undertaking a survey onthe bus service(s) (or lack thereof) in a city or town. If a judgment was made to selectthe sample from only those who used the service(s), this could result in an entirely dif-ferent outcome from what might occur if there had been a balance from both bus-usersand non bus-users. Consequently, we should be wary of surveys conducted using thistechnique (although we probably would not be aware that this technique was themethod used). It is timely to reinforce the fact that, when bombarded with statisticalfacts (and this occurs in our lives daily) we should not accept these figures withoutquestion.

Stratified samplingWhen a sample is selected from apopulation consisting of variousstrata, or levels, it is important to havethe strata or levels in the sample occur-ring in the same proportions as they doin the population.

Consider the situation where a stu-dent council body is to be formed fromYears 8 to 12 students in a school. Itwould not seem fair to have an equalnumber from each of the year levels ifthere were, for instance, twice as manyYear 9 students as there were Year 12students.



Cluster samplingThis method involves selecting clusters within a population and selecting a samplefrom within these clusters. The subgroups selected from the population should be iden-tified. Consider the situation where a market survey is to be conducted on the cost oftertiary education. If the chosen clusters included only ones situated in poorer areas ofthe community, the results of the survey would differ vastly from those occurring fromcluster groups consisting of only ones from more affluent areas. It is important,

The number of students in a school is shown in the table.

A student council is to be formed, consisting of 15 members of the student body. The composition of the council must reflect the proportions in the population. How many from each year level should be chosen?

THINK WRITE

Find the total number of students.

Total number of students = 750

Determine the proportion in each year level.

Proportion of students:Year 12s =

Year 11s =

Year 10s =

Year 9s =

Year 8s =

Multiply these proportions by the sample number and construct a table.Round if necessary.

Check to ensure that the total sample number is 15.

Write the answer. Two students should be chosen from Year 12, two from Year 11, four from Year 10, three from Year 9 and four from Year 8.

Year level Number of students

12 85

11 120

10 180

9 165

8 200

1

285750---------

120750---------

180750---------

165750---------

200750---------

3Yearlevel

Number ofstudents

Number insample

12 85 × 15 = 1.7; i.e. 2

11 120 × 15 = 2.4; i.e. 2

10 180 × 15 = 3.6; i.e. 4

9 165 × 15 = 3.3; i.e. 3

8 200 × 15 = 4;.0 i.e. 4

Total 750 15

85750---------

120750---------

180750---------

165750---------

200750---------

4

5

6

3WORKEDExample



therefore, if using this method of sampling, that the clusters are chosen to be as closelyrepresentative of the population as possible. If this is the case, the survey can yieldquite reliable results in a far shorter time and at a greatly reduced cost when comparedwith collecting data from the whole population.

Capture–recapture samplingCapture–recapture sampling is particularly useful for estimating populations of itemsthat are difficult or impossible to count, such as plants and animals. It is often necessaryto monitor wildlife numbers to prevent the occurrence of plagues and the extinction ofspecies. The technique used is to capture a certain number of the species, tag them,then release them. At a later date, another sample is caught and the number of taggedspecimens in the sample observed. From this information, the population of the speciescan be estimated. This method is best illustrated with a practical example.

Capture–recapture techniquePlace a large number of different coloured disks in a container (without counting the number of each colour or the total). The various colours can represent the variety of wildlife (say, fish in a dam). We are interested in determining the number of yellow-belly fish in the dam (represented by the red-coloured disks in the container).1 Mix the disks thoroughly.2 Draw out two handfuls of disks; mark the red disks to identify them as being

tagged. Count the number of tagged red disks and let this number be t.3 Replace all these disks in the container; mix well.4 Draw out a handful of disks; count the number of red disks in the sample (rs);

count the number of tagged red disks in the sample (ts). Replace all the disks in the container and mix well.

5 Repeat the process of drawing out a handful of disks, counting the number of red disks and the number of tagged red disks in each sample. Continue until data for ten (10) samples have been obtained.

6 Draw up the table below to collate the sample data.

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Trial Number red tagged (ts) Number red (rs)

1

2

3

4

5

6

7

8

9

10

Total Σ ts = Σrs =



7 For a sufficiently large number of trials, we could say that these samples represent the population in miniature. This means that the ratio of tagged red disks in the samples to the number of red disks in the samples should be close to the ratio of the total number of red tagged disks in the population to the total number of red disks (r).

Substitute the three known values in the equation; solve to determine the value of ‘r’. (For a capture–recapture of tagged yellow-belly, r would then represent an estimate for the number of yellow-belly fish in the dam.)

8 Count the number of red disks in the container. How close was this number to your estimate, calculated above?

This method of sampling and population estimation obviously has its limitations. For the yellow-belly example, we are assuming that the situation in the lake remains relatively stable; that the types of fish are uniformly distributed throughout the lake; that the number of births is roughly equal to the number of deaths and that intensive fishing has not occurred in the lake over the time period between tagging and recapture. Any calculations of populations of species in the wild must be considered as estimates, as we can not be certain of exact numbers.

ts∑rs∑

--------- tr--=

In estimating the number of fish in a lake, 500 fish are caught, tagged then released back into the dam. A week later a batch of 80 fish are caught and 25 of them are found to be tagged. Estimate the number of fish in the dam.

THINK WRITE

The proportion of tagged fish in the population closely resembles the proportion of tagged fish in the sample caught later.

=

Form an equation. =

Solve the equation. 25p = 500 × 80

p =

= 1600Write the estimate. There are an estimated 1600 fish in the dam.

1Number tagged fishPopulation of fish

----------------------------------------------- Number tagged in sampleSize of sample

-------------------------------------------------------------

2500

p--------- 25

80------

3

500 80×25

---------------------

4

4WORKEDExample



ABS interviewer surveyThe ABS conducts a census every five years. To monitor changes that might occur between these times, surveys are conducted on samples of the population. The ABS selects a representative sample of the population and interviewers are allocated particular households. It is important that no substitutes occur in the sampling. The interviewer must persevere until the selected household supplies the information requested. It is a legal requirement that selected households cooperate.

The following questionnaire is reproduced from the ABS website <www.jaconline.com.au/maths/weblinks>. It illustrates the format and types of questions asked by an interviewer collecting data regarding employment from a sample.

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MINIMUM SET OF QUESTIONS WHEN INTERVIEWER USED — Q1 to Q17

Q.1. I WOULD LIKE TO ASK ABOUT LAST WEEK, THAT IS, THE WEEK STARTING MONDAY THE … AND ENDING (LAST SUNDAY THE …/YESTERDAY).

Q.2. LAST WEEK DID … DO ANY WORK AT ALL IN A JOB, BUSINESS OR FARM?YesNoPermanently unable to workPermanently not intending to work (if aged 65+ only)

K Go to Q.5K K No More QuestionsK No More Questions

Q.3. LAST WEEK DID … DO ANY WORK WITHOUT PAY IN A FAMILY BUSINESS?

YesNoPermanently not intending to work (if aged 65+ only)

K Go to Q.5K K No More Questions

Q.4. DID … HAVE A JOB, BUSINESS OR FARM THAT … WAS AWAY FROM BECAUSE OF HOLIDAYS, SICKNESS OR ANY OTHER REASON?

YesNoPermanently not intending to work (if aged 65+ only)

K K Go to Q.13K No More Questions

Q.5. DID … HAVE MORE THAN ONE JOB OR BUSINESS LAST WEEK?YesNo

K K Go to Q.7

Q.6. THE NEXT FEW QUESTIONS ARE ABOUT THE JOB OR BUSINESS IN WHICH … USUALLY WORKS THE MOST HOURS.

Q.7. DOES … WORK FOR AN EMPLOYER, OR IN … OWN BUSINESS?EmployerOwn businessOther/Uncertain

K K Go to Q.10K Go to Q.9

Q.8. IS … PAID A WAGE OR SALARY, OR SOME OTHER FORM OF PAYMENT?Wage/SalaryOther/Uncertain

K Go to Q.12K




Q.9. WHAT ARE … (WORKING/PAYMENT) ARRANGEMENTS?Unpaid voluntary workContractor/SubcontractorOwn business/PartnershipCommission onlyCommission with retainerIn a family business without payPayment in kindPaid by the piece/item producedWage/salary earnerOther

K Go to Q.13K K K K Go to Q.12K Go to Q.12K Go to Q.12K Go to Q.12K Go to Q.12K Go to Q.12

Q.10. DOES … HAVE EMPLOYEES (IN THAT BUSINESS)?YesNo

K K

Q.11. IS THAT BUSINESS INCORPORATED?YesNo

K K

Q.12. HOW MANY HOURS DOES … USUALLY WORK EACH WEEK IN (THAT JOB/THAT BUSINESS/ALL … JOBS)?

1 hour or moreLess than 1 hour/no hours

K No More QuestionsK

Insert occupation questions if requiredInsert industry questions if required

Q.13. AT ANY TIME DURING THE LAST 4 WEEKS HAS … BEEN LOOKING FOR FULL-TIME OR PART-TIME WORK?

Yes, full-time workYes, part-time workNo

K K K No More Questions

Q.14. AT ANY TIME IN THE LAST 4 WEEKS HAS …Written, phoned or applied in person to an employer for work?Answered an advertisement for a job?Looked in newspapers?

YesNo

Checked factory notice boards, or used the touchscreens at Centrelink offices?

K

K

K K K

AT ANY TIME IN THE LAST 4 WEEKS HAS …Been registered with Centrelink as a jobseeker?Checked or registered with an employment agency?Done anything else to find a job?Advertised or tendered for workContacted friends/relativesOtherOnly looked in newspapersNone of these

K

K K K K K No More QuestionsK No More QuestionsK No More Questions

Q.15. IF … HAD FOUND A JOB COULD … HAVE STARTED WORK LAST WEEK?YesNoDon’t know

K K K



Reading the questionnaire carefully you will note that, although the questions are labelled 1 to 17, there are only fifteen (15) questions requiring answers (two are introductory statements to be read by the interviewer). Because of directions to forward questions, no individual would be asked all fifteen questions.

1 How many questions would be asked of those who have a job?

2 How many questions would unemployed individuals answer?

3 How many questions apply to those not in the labour force?

Choose a topic of interest to you and conduct a survey

1 Design an interview questionnaire of a similar format to the ABS survey, using directions to forward questions.

2 Decide on a technique to select a representative sample of the students in your class.

3 Administer your questionnaire to this sample.

4 Collate your results.

5 Draw conclusions from your results.

6 Prepare a report which details the:a aim of your surveyb design of the surveyc sample selection techniqued results of the survey collated in table formate conclusions.

Remaining questions are only required if Duration ofUnemployment is neededfor output or to derive the long term unemployed.

Q.16. WHEN DID … BEGIN LOOKING FOR WORK?Enter DateLess than 2 years ago

2 years or more ago

5 years or more ago

Did not look for work

......./......./.......DD MM YY......./......./.......DD MM YY......./......./.......DD MM YY

K

Q.17. WHEN DID … LAST WORK FOR TWO WEEKS OR MORE?Enter DateLess than 2 years ago

2 years or more ago

5 years or more ago

Has never worked (for two weeks or more)

......./......./.......DD MM YY......./......./.......DD MM YY......./......./.......DD MM YY

K No More Questions



Samples and sampling

1 Use the two-digit random number table on page 173. Start at the bottom left-hand cornerthen snake up and down the columns selecting 10 numbers in the range 50 to 99.

2 Use your calculator to generate 10 random integers in the range 50 to 99.

3 Use your calculator to generate a set of random two-digit integers in the range 01 to99. Write these numbers in table format. Use your table (and some random selectiontechnique) to select 10 random integers in the range 50 to 99.

4 Compare your answers to questions 1, 2 and 3. Does it appear that three different setsof random numbers resulted?

5 Describe the techniques employed to select samples in each of the following situations.a Drawing student numbers from a hat to select those to attend the athletics carnivalb Choosing the best student in each class to form a student council bodyc Interviewing the students at the school tuck-shop for an opinion regarding the

school uniformd Selecting those students in a classroom sitting next to a window to form a debating

groupe Selecting one quarter of the students from each year level to represent the school

at a local function

6 For each of the following, state whether the sample used is an example of random,stratified or systematic sampling.a Every tenth tyre coming off a production line is tested for quality.b A company employs 300 men and 450 women. The sample of employees chosen

for a survey contains 20 men and 30 women.c The police breathalyse the driver of every red car.d The names of the participants in a survey are drawn from a hat.e Fans at a football match fill in a questionnaire. The ground contains 8000 grand-

stand seats and 20 000 general admission seats. The questionnaire is then given to40 people in the grandstand and 100 people who paid for a general admission seat.

rememberThere are many methods for selecting a sample. Some important methods include:1. Random sample — chance is the only factor in deciding who is surveyed. This

is best done using a random number generator.2. Stratified sample — those sampled are chosen in proportion to the entire

population.3. Systematic sample — a system is used to choose those who are to be in the

sample.4. Accessibility sample — those within easy access form the sample.5. Quota sample — a quota is imposed on the number in the sample.6. Judgmental sample — judgment is made regarding those to be sampled.7. Cluster sample — those in the sample are chosen from clusters within the

population.8. Capture–recapture sample — used to estimate wildlife populations.

remember

4BWORKEDExample

2



7

Which of the following is an example of a systematic sample?A The first 20 students who arrive at school each day participate in the survey.B Twenty students to participate in the survey are chosen by a random number

generator.C Twenty students to participate in the survey are selected in proportion to the

number of students in each school year.D Ten boys and 10 girls are chosen to participate in the survey.

8

Which of the following statistical investigations would be practical to complete bycensus?A A newspaper wants to know public opinion on a political issue.B A local council wants to know if a skateboard ramp would be popular with young

people in the area.C An author wants a cricket player’s statistics for a book being written.D An advertising agency wants to know the most watched program on television.

9 The table below shows the number of students in each year at a NSW secondaryschool.

If a survey is to be given to 50 students at the school, how many from each Yearshould be chosen if a stratified sample is used?

10 A company employs 300 men and 200 women. If a survey of 60 employees using astratified sample is completed, how many people of each sex participated?

11 The table below shows the age and gender of the staff of a corporation.

A survey of 50 employees is to be done. Using a stratified survey, suggest thebreakdown of people to participate in terms of age and sex.

12 The fish population of a river is to be estimated. A sample of 400 fish are caught,tagged and released. The next day another sample of 400 fish are caught and 40 ofthem have tags. Estimate the fish population of the river.

13 A colony of bats live near a school. Wildlife officers try to estimate the bat populationby catching 60 bats and tagging them. These bats are then released and another 60 arecaught, 9 of which had tags. Estimate the size of the bat population living near theschool.

Year 7 8 9 10 11 12 Total

No. of students 90 110 90 80 70 60 500

Age Male Female

20–29 61 44

30–39 40 50

40–49 74 16

50–59 5 10

mmultiple choiceultiple choice


WORKEDExample

3

4.1 WORKEDExample

4


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 18714 A river’s fish population is to be estimated. On one day 1000 fish are caught, tagged

and released. The next day another 1000 fish are caught. Estimate the population ofthe river if in the second sample of fish:

15 A certain fish population is said to be endangered if the population falls below 15 000.A sample of 1000 fish are caught, tagged and released. The next day another sampleof 1200 fish are caught, 60 of which had tags. Is the fish population endangered?

16 To estimate the population of a lake, 300 fish werecaught. These 300 fish (150 trout, 100 bream and 50perch) were tagged and released. A second sample of fishwere then caught. Of 100 trout, 24 had tags; of 100bream, 20 had tags; and of 100 perch, 8 had tags.a Estimate the number of trout in the lake.b Estimate the number of bream in the lake.c Estimate the number of perch in the lake.

17 The kangaroo population in a national park is to be estimated. On one day, 100 kanga-roos were caught and tagged before being released. (Note: For each sample taken, thekangaroos are released after the number with tags is counted.)a The next day 100 were caught, 12 of which had tags. Estimate the population.b The following day another estimate was done. This time 200 were caught and 20

had tags. Estimate the population again.c A third estimate was done by catching 150 and this time 17 had tags. What will

the third estimate for the population be?d For a report, the average of the three estimates is taken. Calculate this average.

For each of the following (1 to 3), state whether a census or survey has been used.

1 A school votes to elect a school captain.

2 Five hundred drivers complete a questionnaire on the state of a major highway.

3 All insurance customers complete a questionnaire when renewing their policies.

For each of the following (4 to 10), state the type of sample that has been taken.

4 A computer selects 500 phone numbers.

5 Every one thousandth person in the telephone book is selected.

6 Private and business telephone numbers are chosen in proportion to the number of private and business listings.

7 Residents from three suburbs are selected from a town.

8 The best runners from each year level are selected.

9 Twenty students are chosen from a class.

10 All the student numbers are placed in a hat then a sample is chosen.

a 100 had tags b 40 had tags c 273 had tags.

1



BiasNo doubt you have heard the comment, ‘There are lies, damned lies and statistics’. Thisimplies that we should be wary of statistical figures quoted. Indeed, we should alwaysmake informed decisions of our own and not simply accept the mass of statistics thatbombards us through the media.

Bias can be introduced into statistics by:1. questionnaire design2. sampling bias3. the interpretation of results.

Bias in questionnaire designConsider a survey designed to collectdata on opinions relating to cullingkangaroo numbers in Australia.

The questions may be designed to beemotive in nature. Respondents in thesesituations feel obliged to show com-passion. Posing a question in the form,‘The kangaroo is identified as a nativeAustralian animal, not found anywhereelse in the world. Would you be infavour of culling kangaroos in Aus-tralia?’, would almost certainlyencourage a negative response.

Using a leading question (one whichleads the respondent to answer in a par-ticular way) can cause bias to creep intoresponses. Rephrasing the question inthe form, ‘As you know, kangarooscause massive damage on many farmingproperties. You’d agree that theirnumbers need culling, wouldn’t you?’,would encourage a positive response.

Using terminology that is unfamiliarto a large proportion of those being sur-veyed would certainly produce unre-liable responses. ‘Do you think we needto cull herbivorous marsupial mammalsin Australia?’, would cause most respondents to answer according to their under-standing of the terms used. If the survey was conducted by an interviewer, the termcould be explained. In the case of a self-administered survey, there would be no indi-cation of whether the question was understood or not.

Sampling biasAs discussed previously, an ideal sample should reflect the characteristics of the popu-lation. Statistical calculations performed on the sample would then be a reliable indi-cation of the population’s features.

Selecting a sample using a non-random method, as discussed earlier, generally tendsto introduce an element of bias.


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 189Particular responses can be selected from all those received. In collecting infor-

mation on a local issue, an interviewer on a street corner may record responses frommany passers-by. From all the data collected, a sample could be chosen to support theissue, or alternatively another sample could be chosen to refute the same issue.

A sample may be selected under abnormal conditions. Consider a survey to deter-mine which lemonade was more popular – Kirks or Schweppes. Collecting data oneweek when one of the brands was on special at half price would certainly produce mis-leading results.

Data are often collected by radio and television stations via telephone polls. A ‘Yes’response is recorded on a given phone-in number, while the ‘No’ respondents are askedto ring a different phone-in number. This type of sampling does not produce a represen-tative sample of the population. Only those who are highly motivated tend to ring andthere is no monitoring of the number of times a person might call, recording multiplevotes.

When data are collected from mailing surveys, bias results if the non-response rate ishigh (even if the selected sample was a random one). The responses received often rep-resent only those with strong views on the subject, while those with more moderateviews tend to lack representation in their correct proportion.

Statistical interpretation biasOnce the data have been collected, collated and subjected to statistical calculations,bias may still occur in the interpretation of the results.

Misleading graphs can be drawn leading to a biased interpretation of the data.Graphical representations of a set of data can give a visual impression of ‘little change’or ‘major change’ depending on the scales used on the axes (we learned about mis-leading graphs in Year 11).

The use of terms such as ‘majority’, ‘almost all’ and ‘most’ are open to interpret-ation. When we consider that 50.1% ‘for’ and 49.9% ‘against’ represents a ‘majorityfor’ an issue, the true figures have been hidden behind words with very broad mean-ings. Although we would probably not learn the real facts, we should be wary of stat-istical issues quoted in such terms.

Bias in statisticsThe aim of this investigation is to study statistical data that you suspect to be biased.

Conduct a search of newspapers, magazines or any printed material to collect instances of quoted statistics that you believe to be biased. There are occasions when television advertisements quote statistical figures as a result of questionable sampling techniques. For each example, discuss:

1 the purpose of the survey

2 how the data might have been collected

3 the question(s) that may have been asked (try to pose the question(s) in a variety of ways to influence different outcomes)

4 ways in which bias might be introduced

5 variations in interpretation of the data.

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190


Biased sampling

Discuss the problem that would be caused by each of the following biased samples.

1

A survey is to be conducted to decide the most popular sport in a local community. A sample of 100 people was questioned at a local football match.

2

A music store situated in a shopping centre wants to know the type of music that it should stock. A sample of 100 people was surveyed. The sample was taken from people who passed by the store between 10 and 11 am on a Tuesday.

3

A newspaper conducting a Gallup poll on an election took a sample of 1000 people from the Gold Coast.

Spreadsheets creating misleading graphs

We looked at creating misleading graphs in the Year 11 text. Let us practise that investigation again to reinforce the techniques used to produce misleading graphs. Consider the data in this table.

We shall use a spreadsheet to produce misleading graphs based on these data.

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vestigation

investigation

Year 1985 1990 1995 2000

Wages% increase in wagesProfits% increase in profits

625

120

950

1·550

1344

2·566

2054

5100

Graph 2

Graph 1

Graph 3

Maths A Yr 12 - Ch. 04 Page 190 Thursday, September 12, 2002 11:09 AM


1 Enter the data as indicated in the spreadsheet (see page 190).2 Graph the data using the Chart Wizard. You should obtain a graph similar to

Graph 1.3 Copy and paste the graph twice within the spreadsheet.4 Graph 2 gives the impression that the wages are a great deal higher than the

profits. This effect was obtained by reducing the horizontal axis. Experiment with shortening the horizontal length and lengthening the vertical axis.

5 In Graph 3 we get the impression that the wages and profits are not very different. This effect was obtained by lengthening the horizontal axis and shortening the vertical axis. Experiment with various combinations.

6 Print out your three graphs and examine their differences.Note that all three graphs have been drawn from the same data using valid scales. A cursory glance leaves us with three different impressions. Clearly, it is important to look carefully at the scales on the axes of graphs.

Another method which could be used to change the shape of a graph is to change the scale of the axes.7 Right click on the axis value, enter the Format axis option, click on the Scale

tab, then experiment with changing the scale values on both axes.Techniques such as these are used to create different visual impressions of the same data.8 Use the data in the table to create a spreadsheet, then produce two graphs

depicting the percentage increase in both wages and profits over the years giving the impression that:a the profits of the company have not grown at the expense of wage increases

(the percentage increase in wages is similar to the percentage increase in profits)b the company appears to be exploiting its employees (the percentage

increase in profits is greater than that for wages).

Discuss why the following selected samples could provide bias in the statistics collected.a In order to determine the extent of unemployment in a community, a committee phoned two

households (randomly selected) from each page of the local telephone book during the day.b A newspaper ran a feature article on the use of animals to test cosmetics. A form

beneath the article invited responses to the article.

Continued over page

THINK WRITE

a Consider phone book selection. a Phoning two randomly selected households per page of the telephone directory is possibly a representative sample.

Consider those with no phone contact.

However, those without a home phone and those with unlisted numbers could not form part of the sample.

Consider the hours of contact. An unanswered call during the day would not necessarily imply that the resident was at work.

1

2

3

5WORKEDExample



Bias

1 Rewrite the following questions, removing any elements or words that might contributeto bias in responses.a The poor homeless people, through no fault of their own, experience great hardship

during the freezing winter months. Would you contribute to a fund to build a shelterto house our homeless?

b Most people think that, since we’ve developed as a nation in our own right andbroken many ties with Great Britain, we should adopt our own national flag. You’dagree with this, wouldn’t you?

c You’d know that our Australian 50 cent coin is in the shape of a dodecagon,wouldn’t you?

d Many in the workforce toil long hours for low wages. By comparison, politiciansseem to get life pretty easy when you take into account that they only work for partof the year and they receive all those perks and allowances. You’d agree, wouldn’tyou?

2 Rewrite parts a to d in question 1 so that the expected response is reversed.

3 What forms of sampling bias can you identify in the following samples?a Choosing a sample from students on a bus travelling to a sporting venue to answer

a questionnaire regarding sporting facilities at their schoolb Sampling using ‘phone-in’ responses to an issue viewed on a television program.c Promoting the results of a mail-response survey when fewer than half the selected

sample replied.d Comparing the popularity of particular chocolate brands when one brand has a ‘two

for the price of one’ special offer.e Choosing a Year 8 class and a Year 12 class to gather data relating to the use of the

athletics oval after school.

THINK WRITE

b Consider the newspaper circulation. b Selecting a sample from a circulated news-paper excludes those who do not have access to the paper.

Consider the urge to respond. In emotive issues such as these, only those with strong views will bother to respond, so the sample will represent extreme points of view.

1

2

rememberBias can be introduced at each of the following stages:1. questionnaire design2. sampling bias3. interpretation of results.

remember

4C

WORKEDExample

5


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 1934 Why does this graph produce a biased visual impression?

5 Comment on the following statement:‘University tests have demonstrated that Double-White toothpaste is consistently usedby the majority of teenagers and is more effective than most other toothpastes.’

6 Surveys are conducted on samples to determine the characteristics of the population.Discuss whether the samples selected would provide a reliable indication of the popu-lation’s characteristics.

Sample Populationa Year 11 students Student driversb Year 12 students Students with part-time jobsc Residents attending a Residents of a suburb

neighbourhood watch meetingd Students in the school choir Music students in the schoole Cars in a shopping centre car park Models of Holden cars on the roadf Males at a football match Popular TV programsg Users of the local library Popular teenage magazines

BiasIt is important that a sample is chosen randomly to avoid bias. Consider the following situation.

The government wants to improve sporting facilities in Brisbane. They decide to survey 1000 people about what facilities they would like to see improved. To do this, they choose the first 1000 people through the gate at a football match at Lang Park.

In this situation it is likely that the results will be biased towards improving facilities for football. It is also unlikely that the survey will be representative of the whole population in terms of equality between men and women, age of the participants and ethnic backgrounds.Questions can also create bias. Consider asking the question, ‘Is football your

favourite sport?’ The question invites the response that football is the favourite sport rather than allowing a free choice from a variety of sports by the respondent.

Consider each of the following surveys and discuss:a any advantages, disadvantages and possible causes of biasb a way in which a truly representative sample could be obtained.

Value of A$ compared with US $1

Aus

tral

ian

curr

ency 51c

50c

49c

9 May 11 May 12 MayDate

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Contingency tablesWhen sample data are collected, it is often useful to break the data into categories. Atwo-way frequency table or contingency table displays data that have been classifiedinto different types.

Consider, for example, data collected on the hair colour of 200 couples. It may berepresented in a table such as the one below.

These data could be represented as a 3-dimensional bar chart, as shown below.

1 Surveying food product choices by interviewing customers of a large supermarket chain as they emerge from the store between 9.00 am and 2.00 pm on a Wednesday.

2 Researching the popularity of a government decision by stopping people at random in a central city mall.

3 Using a telephone survey of 500 people selected at random from the phone book to find if all Australian States should have Daylight Saving Time in summer.

4 A bookseller uses a public library database to survey for the most popular novels over the last three months.

5 An interview survey about violence in sport taken at a rugby league football venue as spectators leave.

Female

Red Dark Fair Total

Male

Fair 11 25 9 45

Dark 19 51 28 98

Red 17 27 13 57

Total 47 103 50 200Fair

Fair

DarkDarkRed

Red

0102030405060

Hair colour of 200 couples

Female hair colour Mal

e ha

ir co

lour

Freq

uenc

y


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 195Although this graph displays the data so that comparisons are readily visible, the

chart is difficult to read and figures can not be read accurately.If we considered representing the data as a 2-dimensional segmented bar chart, this

could be done in two ways.Splitting the data into categories based on the hair colour of the male and calculating

percentages in each category would yield the following figures and segmented bargraph:

Splitting the data into categories based on the hair colour of the female and calcu-lating percentages in each category would yield the following figures and segmentedbar graph:

It is obvious that the interpretation of thedata depends on the reference basis. We maywish to interview those couples where themale is fair haired and the female darkhaired. Note that this represents 25 couples.What if we talk about percentages? Com-paring the percentages in the two tables, itcan be seen that:1. 56% of fair-haired males have female

partners with dark hair2. 24% of dark-haired females have male

partners with fair hair.These percentages have vastly differentvalues, yet they both describe the same set of25 couples of fair-haired males and dark-haired females. It is important, particularlywhen dealing with contingency tables, toconsider the reference basis for percentages.

Female

Red Dark Fair Total

Male

Fair 24% 56% 20% 100%

Dark 19% 52% 29% 100%

Red 30% 47% 23% 100%

Female

Red Dark Fair

Male

Fair 23% 24% 18%

Dark 41% 50% 56%

Red 36% 26% 26%

Total 100% 100% 100%

FairDarkRed

Hair colour of male

0

20%

40%

60%

80%

100%Hair colour of 200 couples

Red

FairDark

Hair colour of female

Fair

Dark

Red

Hai

r co

lour

of

mal

e

0 20% 40% 60% 80%100%

Hair colour of 200 couples

Red

FairDark

Hair colour of female



When information on a test is presented in a contingency table, conclusions can bemade about the accuracy of the test.

A new test was designed to assess the reading ability of students entering high school. The results were used to determine if the students’ reading level was adequate to cope with high school. The students’ results were then checked against existing records.Of the 150 adequate readers who sat for the test, 147 of them passed.Of the 50 inadequate readers who sat for the test, 9 of them passed.Present this information in a contingency table.

THINK WRITE

Draw up the table showing the number of students whose reading was adequate and the number of students for whom the results of the new test were confirmed.

Test results

TotalPassedDid not

pass

Adequate readers

147 3 150

Inadequate readers

9 41 50

Total 156 44

6WORKEDExample

A batch of sniffer dogs is trained by customs to smell drugs in suitcases. Before they are used at airports they must pass a test. The results of that test are shown in the contingency table below.

a How many bags did the sniffer dogs examine?b In how many bags did the dogs detect drugs?c In what percentage of bags without drugs did the dogs incorrectly detect drugs?d Based on the above results, what percentage of the time will the dogs not detect a bag

carrying drugs?

Test results

TotalDetected Not detected

No of bags with drugs 24 1 25

No. of bags without drugs 11 164 175

Total 35 165

7WORKEDExample



As a result of studying a contingency table, we should also be able to make judge-ments about the information given in the tables. In the previous worked exampleonly one bag out of 25 with drugs went undetected. Although the dogs incorrectlydetected drugs in 11 bags that did not have drugs, they still have an overall accu-racy of 94% as shown by the calculation [(24 + 164) ÷ 200] × 100%.

Many contingency tables will require you to make your own value judgements aboutthe conclusions established. For example, the 94% overall accuracy recorded may beconsidered ‘very acceptable’.

THINK WRITE

a Add both total columns; they should give the same result.

a 200 bags were examined.

b The total of the detected column. b The dogs detected drugs in 35 bags.

c There were 175 bags without drugs but dogs incorrectly detected them in 11 bags. Write this as a percentage.

c Percentage incorrectly detected

= × 100%

= 6.3%

d Of 25 bags with drugs, 1 went undetected. Write this as a percentage.

d Percentage not detected = × 100%Percentage not detected = 4%

11175---------

125------

The contingency table at right shows the composition of the employees of a small law firm.a Extend the table to show totals in all

categories and an overall total.b Draw a table showing percentages with

respect to type of employment (full or part-time).c Redraw the table showing percentages based on the gender of the employee.d What percentage of females work full time?e What percentage of full-time workers are female?f Explain why, in the workforce in general, it would be easier to estimate an answer to

part d than it would to obtain an estimate for part e.

Continued over page

THINK WRITE

a Add the numbers in the cells for all the rows and columns and enter the totals. Check that the overall total is consistent for the rows and columns.

Full-time Part-time

Female 4 11

Male 30 5

aFull-time Part-time Total

Female 4 11 15

Male 30 5 35

Total 34 16 50

8WORKEDExample



Climatic influences in QueenslandFor this activity we will investigate relationships between geographical features that influence our weather. We could pose questions such as:

What effect does latitude have on temperature?What factor has the main influence on day length?What part does elevation play in influencing temperature?

This investigation should be conducted using a spreadsheet. Data on Queensland towns from the Bureau of Meteorology’s website have been collated and shown in two spreadsheet tables which follow. Graphs have been provided for stimulus when investigating relationships between the variables in the spreadsheet.

1 Retrieve the two Excel files from the CD provided with this book (the longitude file does not contain the graphs displayed here).

2 Experiment by graphing pairs of variables to determine whether a relationship exists between the pair. You may wish to sort the spreadsheet using a different classification.

THINK WRITE

b Percentages are based on totals in columns. The totals in the columns are on the denominator when calculating percentages.

c Percentages are based on totals in rows. The totals in the rows are on the denominator when calculating percentages.

d This is based on female totals in table c.

d × 100 = × 100 = 27%

Write the answer. Percentage of females who work full time = 27%.

e This is based on full-time totals in table b.

e × 100 = × 100 = 12%

Write the answer. Percentage of full-time workers who are female = 12%.

f An estimate is easier if the required sample is smaller.

f It would be easier to obtain an estimate for the percentage of females who work full time because the number of females is fewer than the number of full-time workers. This means that the sample size would be smaller.

bFull-time Part-time

Female × 100 = 12% × 100 = 69%

Male × 100 = 88% × 100 = 31%

Total 100% 100%

434------ 11

16------

3034------ 5

16------

cFull-time Part-time Total

Female × 100 = 27% × 100 = 73% 100%

Male × 100 = 86% × 100 = 14% 100%

415------ 11

15------

3035------ 5

35------

1Full time

Female total----------------------------- 4

15------

2

1Female

Full-time total---------------------------------- 4

34------

2

Locality by latitude

Locality by longitude

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3 Write a report on the geographical factors influencing daily temperatures and sunlight hours. Support your conclusions by providing graphical evidence.

4 Sites on the World Wide Web provide weather conditions for many places throughout the world. Conduct a search to collate data from locations around the globe. Investigate the geographical features which might have an influence on their weather.



We are constantly bombarded with statistics, some of which are a valid interpretation ofthe data, and some of which are not. On occasions, the misuse of statistics may beunintentional or through ignorance, but there are occasions when misleading figures arequoted intentionally. If the raw data are available, it is wise to check the validity of anyclaims.

The ABS data from the 1996 Census for the Chapel Hill area in Brisbane are shown here.Note: Income figures are weekly income expressed in AUD.

Australian Bureau of Statistics1996 Census of Population and HousingChapel Hill (Statistical Local Area) — Queensland

B01 Selected Characteristics — Chapel Hill

Male Female PersonsTotal persons (a) 4 824 5 112 9 936Aged 15 years and over (a) 3 761 4 070 7 831Aboriginal 3 3 6Torres Strait Islander 0 0 0Both Aboriginal and Torres Strait Islander (b) 0 0 0Australian born 3 405 3 704 7 109Born overseas: Canada, Ireland, NZ, South Africa, UK (c) and USA 696 637 1 333Born overseas: Other country (d) 587 605 1 192Born overseas: Total 1 283 1 242 2 525Speaks English only and aged 5 years and over 3 936 4 230 8 166Speaks language other than English (e) and aged 5 years and over 459 491 950Australian citizen 4 239 4 515 8 754Australian citizen aged 18 years and over 3 027 3 315 6 342Unemployed 141 132 273Employed 2 677 2 427 5 104In the labour force 2 818 2 559 5 377Not in the labour force 854 1 393 2 247Enumerated in private dwelling (a) 4 815 5 085 9 900Enumerated in non-private dwelling (a) 9 27 36Persons enumerated same address 5 years ago 2 217 2 381 4 598Persons enumerated different address 5 years ago 2 163 2 324 4 487Overseas visitor 54 75 129

Chapel HillMedian age 34Median individual income 415Median household income 1 209Average household size 3.1

When discussing the probability of unemployment in this area, a resident proudly saidthat only 5% of the unemployed in the area were male.a Construct a contingency table displaying the employment/unemployment status of the

residents in this area.b Use your contingency table to discuss the validity of the claim.

THINK WRITE

a Extract the employment and unemployment figures for males and females from the table.

a

Form a contingency table adding totals to rows and columns.

1Male Female Total

Unemployed 141 132 273

Employed 2677 2427 5104

Total 2818 2559 5377

2

9WORKEDExample



An error frequently occurs when statistics of this kind are quoted. The reference basisfor the probability percentage should be carefully noted.

When using the Maths Quest Maths A Year 12 CD-ROM, click here for more aboutstatistical measures.

Contingency tables from census dataThe table below displays data collected from the 1996 census. It shows the numbers of males and females in various forms of employment in the 15–19 years age bracket and the totals of all ages for each category.

Australian bureau of statistics 1996 Census of Population and Housing Australia 7688965.464 sq kms

THINK WRITE

b Calculate the probability of an unemployed person being a male; that is,

× 100.

b P(unemployed being male) = × 100

P(unemployed being male) = 52%52% of the unemployed are males.

Calculate the probability of a male being unemployed; that is,

× 100.

P(male being unemployed) = × 100

P(male being unemployed) = 5%5% of the males are unemployed.

Compare these probability figures with the statement and make a decision.

The statement is not correct. The resident should have said that only 5% of the males were unemployed.

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extensioneextensionxtensionMeasures oflocation andspread extensioneextensionxtension Comparison

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15–19 years TotalIndustry Male Female Male Female PersonsAgriculture, Forestry and Fishing 9 986 2 685 225 679 98 651 324 330Mining 1 349 339 75 497 10 764 86 261Manufacturing 33 831 10 168 695 007 270 029 965 036Electricity, Gas and Water Supply 676 191 49 427 9 272 58 699Construction 21 162 1 664 419 394 64 690 484 084Wholesale Trade 11 441 5 367 306 456 140 089 446 545Retail Trade 91 818 127 466 500 105 536 543 1 036 648Accommodation, Cafes and Restaurants 17 917 25 019 157 519 197 768 355 287Transport and Storage 4 135 2 560 250 385 81 693 332 078Communication Services 1 213 869 102 016 48 172 150 188Finance and Insurance 2 001 4 981 127 364 169 092 296 456Property and Business Services 11 164 13 930 410 414 339 781 750 195Government Administration and Defence 5 877 2 999 225 316 148 111 373 427Education 3 685 4 071 184 287 365 776 540 063Health and Community Services 3 059 13 388 161 489 563 689 725 178Cultural and Recreational Services 6 658 7 016 93 066 85 989 179 055Personal and Other Services 4 161 11 212 143 942 133 966 277 908Non-classifiable economic units 3 808 2 000 63 045 40 097 103 142Not stated 9 133 8 175 81 643 70 096 151 739Total 243 074 244 100 4 272 051 3 364 268 7 636 319



Contingency tables

1 A test is developed to test for infection with the flu virus. To test the accuracy, thefollowing 500 people are tested.• Of the 100 people who are known to have the flu who are tested, the test returns 98

positive results.• Of the 400 people who are known not to be infected with the virus who are tested,

12 false positives are returned.

Using these data, we could form a contingency table to compare the proportion of 15–19 year old males and females in, for example, the retail trade. (Confirm the figures in the table below.)

1 Use this table to:a determine the percentage of male workers who are in the retail tradeb calculate the percentage of retail workers who are malec explain why these two percentages are differentd plan a strategy to survey the workforce for an estimate of the number of

males in the retail trade.2 Choose another category of the workforce from the census data. Construct a

contingency table, then answer questions similar to those above.3 Reports from early recordings of census data showed that more than 50% of

Australians lived and worked on the land, providing food and clothing for our population. Most recent reports indicate that only 4% of Australians now work the land, providing for the remaining 96%. Use the data in the table to confirm that this is indeed true.

4 It is important for future planning that these changes are recorded and made known. Search the World Wide Web or reference books to obtain industry data from the 2001 census. Examine the figures, noting changing trends in industry employment. Report on your findings.

Male Female Total

Retail trade 91 818 127 466 219 284

Non-retail trade 151 256 116 634 267 890

Total 243 074 244 100 487 174

remember1. Contingency tables can be used to display data that have been classified into

different types.2. The table displays 2 variables which have been split into categories in a

horizontal and a vertical direction.3. Calculations can be made with regard to a variety of reference bases.

remember

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C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 203Display this information in the contingency table below.

2 One thousand people take a lie detector test. Of 800 people known to be telling thetruth, the lie detector indicates that 23 are lying. Of 200 people known to be lying, thelie detector indicates that 156 are lying.Present this information in a contingency table.

3 The contingency table shown below displays the information gained from a medicaltest screening for a virus. A positive test indicates that the patient has the virus.

a How many patients were screened for the virus?b How many positive tests were recorded? (that is, in how many tests was the virus

detected?)c What percentage of test results were accurate?d Based on the medical results, if a positive test is recorded what is the percentage

chance that you actually have the virus?

4 The contingency table below indicates the results of a radar surveillance system. If thesystem detects an intruder, an alarm is activated.

a Over how many nights was the system tested?b On how many occasions was the alarm activated?c If the alarm is activated, what is the percentage chance that there actually is an

intruder?d If the alarm was not activated, what is the percentage chance that there was an

intruder?e What was the percentage of accurate results over the test period?f Comment on the overall performance of the radar detection system.

Test results

TotalAccurate Not accurate

With virus

Without virus

Total

Test results


With virus 45 3 48

Without virus 922 30 952

Total 967 33 1000

Test results

TotalAlarm activated Not activated

Intruders 40 8 48

No intruders 4 148 152

Total 44 156 200

WORKEDExample

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4.2

4.3



The information below is to be used in questions 5 to 7.A test for a medical disease does not always produce the correct result. A positive testindicates that the patient has the condition. The table indicates the results of a trial on anumber of patients who were known to either have the disease or known not to have thedisease.

5The overall accuracy of the test is:A 90% B 90.5% C 92.5% D 95%

6Based on the table, what is the probability that a patient who has the disease has itdetected by the test?A 90% B 90.5% C 92.5% D 95%

7Which of the following statements is correct?A The test has a greater accuracy with positive tests than with negative tests.B The test has a greater accuracy with negative tests than with positive tests.C The test is equally accurate with positive and negative test results.D There is insufficient information to compare positive and negative test results.

8 Airport scanning equipment is tested by scanning 200 pieces of luggage. Prohibiteditems were placed in 50 bags and the scanning equipment detected 48 of them. Theequipment detected prohibited items in five bags that did not have any forbidden itemsin them.a Use the above information to complete the contingency table below.

b Use the table to answer the following:i What percentage of bags with prohibited items were detected?ii What was the percentage of ‘false positives’ among the bags that had no

prohibited items?iii What percentage of prohibited items pass through the scanning equipment

undetected?iv What is the overall percentage accuracy of the scanning equipment?

Test results


With disease 57 3 60

Without disease 486 54 540

Total 543 57 600

Test results


Bags with prohibited items

Bags with no prohibited items

Total





C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 2059 In some cases it is easier to count numbers in a particular category by considering a

different population. In each of the following pairs of proportions, which one wouldbe easier to determine?a ii Proportion of males who are left-handed.

ii Proportion of left-handers who are males.b ii Proportion of mathematics A students in your school who are over 16.

ii Proportion of over 16 year olds in your school who study mathematics A.c ii Proportion of state school students who live in Queensland.

ii Proportion of Queensland school students who attend a state school.

10 Refer to the 1996 census data on industry on page 201.a Draw up a contingency table showing the 15–19 year old males and females

employed in education compared with those of this age group employed in otherindustries.

b Extend your table to show totals in all categories as well as an overall total.c Draw up a table showing percentages with respect to gender.d Redraw your table showing percentages based on industry.e What percentage of females are employed in education?f What percentage of those employed in education are female?g At some period in between census times, if it were necessary to obtain an estimate

of the number of females employed in education by surveying a sample, whatapproach would you recommend?

11 Repeat question 10 using the ‘totals’ data. Comment on any differences or similaritiesin your answers.

Use the following data collected from the 1996 census for questions 12 and 13.Note: Income figures are weekly income in AUD.

Australian Bureau of Statistics 1996 Census of Population and Housing Inala (Statistical Local Area) — QueenslandB01 Selected Characteristics — Inala

Male Female PersonsTotal persons (a) 6 401 6 886 13 287Aged 15 years and over (a) 4 516 5 083 9 599Aboriginal 398 482 880Torres Strait Islander 42 51 93Both Aboriginal and Torres Strait Islander (b) 8 12 20Australian born 4 066 4 468 8 534Born overseas: Canada, Ireland, NZ, South Africa, UK (c) and USA 665 694 1 359Born overseas: Other country (d) 1 396 1 462 2 858Born overseas: Total 2 061 2 156 4 217Speaks English only and aged 5 years and over 4 000 4 483 8 483Speaks language other than English (e) and aged 5 years and over 1 454 1 494 2 948Australian citizen 5 424 5 807 11 231Australian citizen aged 18 years and over 3 521 4 001 7 522Unemployed 605 349 954Employed 2 036 1 403 3 439In the labour force 2 641 1 752 4 393Not in the labour force 1 728 3 144 4 872Enumerated in private dwelling (a) 6 397 6 886 13 283Enumerated in non-private dwelling (a) 4 0 4Persons enumerated same address 5 years ago 2 986 3 343 6 329Persons enumerated different address 5 years ago 2 319 2 493 4 812Overseas visitor 13 15 28

InalaMedian age 30Median individual income 187Median household income 412Average household size 2.8

WORKEDExample

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WORKEDExample

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12 a Construct a contingency table displaying males and females ‘In the labour force’and ‘Not in the labour force’, showing all totals.

b From your contingency table calculate:i the percentage of females in the labour forceii the percentage of those in the labour force who are female.

c Would it be correct to say that more than 39% of the females are in the labourforce? Explain.

13 a Construct a contingency table displaying the number of ‘Australian born’ and‘Overseas born’ males and females in the community. Show all totals.

b Is it correct to claim that almost half the males in the community were born over-seas? Explain.

Applications of statistics and probability

By exploring data collected from samples (provided the samples have been chosencarefully) we are able to estimate characteristics of the population. We can determinepast trends and speculate on future trends. Through a series of investigations we willexplore the application of statistics and probability to life-related situations.

Using histograms to estimate probabilitiesDiscrete data (the type where the scores can take only set values) can be representedas a frequency histogram.

Continuous data (the type where the scores may take any value, usually within acertain range) can also be represented in the form of a frequency or probability histo-gram. Let us construct a frequency histogram of continuous data from which we canthen estimate probabilities.

4.2

A battery company tested a random sample of a batch of their batteries to determine theirlifetime. The results are shown below.

a Represent the data as a frequency histogram.b If you chose a battery from this batch, estimate the probability that the battery would

last:ii at least 25 hoursii less than 40 hours.

c In an advertising campaign, the battery manufacturer claims that they will replace the battery if it does not last at least 30 hours. Based on these results, what is the probability they will have to replace a battery?

Lifetime (hours) 20–<25 25–<30 30–<35 35–<40 40–<45 45–<50

Frequency 6 25 70 61 30 8

10WORKEDExample



It should be noted that, if we are not given a table of results (as we were in the previousworked example), but simply a frequency histogram, we would have to estimate fre-quencies from the histogram. In this case, the probability answers obtained would beestimates rather than exact values.

THINK WRITE

a Construct a frequency histogram with lifetime on the x-axis and frequency on the y-axis.

a

b Find the total number of scores. b Total number of scores= 6 + 25 + 70 + 61 + 30 + 8= 200

The total area under the curve is 1, so each class interval represents a fraction of 1 in terms of area (and probability).

ii Find the total of frequencies with a score of at least 25 hours.

ii Total frequency at least 25 hours= 25 + 70 + 61 + 30 + 8= 194

Estimated probability

= × 1

P(≥25 h) = × 1P(≥25 h) = 0.97

Write the answer. The probability that the battery would last for at least 25 hours is 0.97.

ii Find the total frequencies with a score of less than 40 hours.

ii Total frequency less than 40 hours= 6 + 25 + 70 + 61= 162

Apply the same rule as in part i. P(<40 h) = × 1P(<40 h) = 0.81

Write the answer. The probability that the battery would last less than 40 hours is 0.81.

c Find the total frequency for those batteries lasting less than 30 hours.

c Total frequency less than 30 hours= 6 + 25= 31

P(<30 h) = × 1= 0.155

Apply the probability rule. P(replacing battery) = 0.155Write the answer. The probability that the manufacturer

will have to replace the battery is 0.155.

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10203040506070

Lifetime (hours)

Frequency histogram

12

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total of frequencies at least 25 htotal number of scores

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194200---------

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2162200---------

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31200---------

23



Interpreting histogramsThe aim of this investigation is to highlight the pitfalls in interpreting the shape of histograms. The activity is more readily conducted using a graphics calculator.

1 Consider the percentages received by a class of 36 students in their end-of-semester test.67, 90, 83, 85, 73, 80, 78, 79, 68, 71, 53, 65, 74, 64, 77, 56, 66, 63, 70, 49, 56, 71, 67, 58, 60, 72, 67, 57, 60, 90, 63, 88, 78, 46, 64, 81.

2 Enter the data as a list into a graphics calculator.

TI: Press and select 1:Edit then enter the data into L1 as shown.

Casio: Enter STAT from the MENU then enter the data into List 1.

3 Set the window for the percentage range 40 to 100 using a class interval of 10.

TI: Press and enter values as shown.

Casio: Press (V-WIN), then enter the values shown.

4 Set the data to graph as a histogram.

TI: Press [STAT PLOT] and select 1:Plot1. Set Plot 1 as shown.

Casio: Press (GRPH) and (SET). Then set StatGraph1

as shown. (Press to scroll right to find Hist.)

5 Set all other graph plots off.

TI: Press [STAT PLOT] and set other plots off.

Casio: Press (SEL) and set other StatGraphs off.

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For more on probability, scatterplots, histograms and skewness, click on the icon whenusing the Maths Quest Maths A year 12 CD-ROM.

6 Draw the histogram.

TI: Press . Casio: Press (DRAW) and enter the values shown.

Press (DRAW).

7 On your calculator, change the range of the score to accommodate percentages 46 to 94, with a class interval of 4.

TI: Press and enter the new values shown.

Casio: Press then press (GPH1) and enter the new

values.

8 Draw the resulting histogram.

TI: Press . Casio: Press (DRAW).

While the first histogram appeared to have one modal class, this one appears multimodal.

9 Use your calculator to investigate changing the class interval and the range of the percentages. What do you observe?

10 All these histograms are graphical representations of the same data. While they all indicate distributions with higher frequencies towards the middle, some suggest bimodal or multimodal distributions. What do you conclude from this investigation?

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Probability, scatterplots,histograms and skewness



Using scatterplots to consider relationships between data sets

Are tall mothers likely to produce tall sons?The table below details the heights of 12 mothers and their adult sons.

a Construct a scatterplot of the data.b Draw the line of best fit.c Estimate the height of a son born to a 180-cm tall mother.d Discuss the relationship between the heights of mothers and their sons as shown by

these data.The solution to this problem will be shown using three methods.1. Pen and paper 2. Graphics calculator 3. SpreadsheetIt should be noted that, when a line of best fit is drawn by eye, variations in answers will occur for those dependent on the position of the line.

Height of mother (cm) 185 152 168 166 173 172 159 154 168 148 162 171

Height of son (cm) 188 162 168 172 179 182 160 148 178 152 184 180

THINK WRITE/DRAW

Method 1. Using pen and papera Plot points on a graph with height of

mother on x-axis (the independent variable) and height of son on y-axis (the dependent variable). This results in a scatterplot.

a

b Draw in the line of best fit. Balance an equal number of points either side of the line and as close to the line as possible.

b

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Continued over page

THINK WRITE/DISPLAY

c Draw a vertical line from the 180 cm point on the x-axis to the line of best fit. From this point on the line, draw a horizontal line to the y-axis. Read this y-value.

c

From the graphwhen x = 180

y = 190So, a 180-cm tall mother could produce a son approximately 190 cm tall.

d Look at the slope of the line and the proximity of the points to the line.

d The slope of the line of best fit is positive, indi-cating that, as one variable increases, the other also increases. The points lie fairly close to the line, so this indicates a fairly strong positive relationship between the two variables. This seems to support the view that tall mothers are likely to produce tall sons.

Method 2. Using a graphics calculatorThese instructions apply to the TI-83 and Casio CFX-9850 PLUS graphics calculators.

TI Casioa Enter mother’s height and son’s

height into two lists.TI: Press , select 1:Edit and enter the data.Casio: From the MENU enter the STAT sector and enter the data.

a

Set up the window to graph x-values in the range 145–190, y-values in the same range and a scale of 5 for each.TI: Press then enter the values shown.Casio: Press (V-WIN) and enter values.

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THINK WRITE/DISPLAYTI Casio

Set one graph plot on and select scatterplot type, x as list 1 and y as list 2.TI: Press [STAT PLOT] and set up Plot 1 as shown.Casio: Press (GRPH),

(SET) and set StatGraph1 as shown.Turn off all other plots. TI: Use [STAT PLOT].Casio: Press (SEL).

Graph the relationship. TI: Press . Casio: Press (DRAW).

b Enter the function that calculates the equation of the line of best fit. TI: Press , arrow across to the CALC menu, and select 4:LinReg. Casio: Press (x).

b

Copy this equation into Y=. TI: Press VARS, select 5:Statistics, arrow across to the EQ menu, and select 1: RegEQ.Casio: Press (COPY) and

to store.Graph the line of best fit on the scatterplot. TI: Press .Casio: Press (DRAW).

c Use the calculator’s function to determine a value for y when the x-value is 180. TI: Press [CALC] and select 1:Value.Casio: Press , select GRAPH, and press (DRAW). Then press

(SLV), and (Y-CAL) and enter 180 value for x and press

.

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THINK WRITE/DISPLAY

d Look at the angle of the straight line and the proximity of the points to the line.

d The line of best fit predicts that a 180-cm tall mother could produce an adult son approximately 187 cm tall. The slope of the line of best fit is upwards, indicating that as one variable increases the other also increases. Most of the points lie close to the line, so it is reasonable to assume that the relationship between mother and son heights is quite strong. This supports the proposal that tall mothers are likely to produce tall sons.

Method 3. Using a spreadsheeta Open up a spreadsheet and enter

the data for the mother’s and son’s heights in columns under headings.

a

Use the chart wizard to graph the data as a scatterplot.Label the axes and provide a title for the graph.Adjust the range and scale on the x- and y-axes to more appropriate values if necessary (suggest 145 to 190 range with a scale of 5).Print out a copy of the scatterplot.

b Draw in the line of best fit. Balance an equal number of points either side of the line and as close to the line as possible.

b From the scatterplot of the data above, the line of best fit is shown on the scatterplot.

c From the graph, read the corresponding y-value for x = 180 cm.

c When x = 180, y = 187. So a 180-cm tall mother would produce an adult son approximately 187 cm tall.

d Look at the slope of the line and the proximity of the points to the line.

d The slope of the line of best fit is positive, indi-cating that, as one variable increases, the other also increases. The points lie fairly close to the line, so this indicates a fairly strong positive relationship between the two variables. This seems to support the view that tall mothers are likely to produce tall sons.

1

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Applications of statistics and probability

1 In the distribution on the right:a is the graph symmetrical?b what is the modal class(es)?c can the mean and median be seen from the

graph and, if so, what are their values?d which score has the greatest probability of

occurring?

2 For the distribution shown on the right:a are the data symmetrical?b what is the modal class(es)?c can the mean and median be seen from the

graph and, if so, what are their values?d which classes have the same probability of

occurring?e which class has the least probability of

occurring?

3 The table on the right shows the number of goals scored by a hockey team throughout a season.a Show this information in a frequency

histogram.b Are the data symmetrical?c What is the mode(s)?d Can the mean and median be seen for this

distribution and, if so, what are their values?

e The probability that the team will score 5 goals is the same as their probability of scoring what other number of goals?

4 For the distribution shown on the right:a what is the modal score(s)?b which score has the greatest probability of occurring?c which score has the least probability of occurring?

remember1. Frequency histograms can be used to estimate probabilities in data sets.2. Scatterplots display the relationship between two variables.3. Scatterplots enable past and future trends to be considered.

remember

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C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 2155 The table at right shows the number of

goals scored by a basketball teamthroughout a season.a Draw a frequency histogram of the

data.b What is the probability that the team

will score more than 40 goals?

6

Which of the distributions below has the smallest standard deviation?A B C D

7 A movie is shown at a cinema 30 times during the week. The number of people attending each session of the movie is shown in the table at right.a Present the data in a frequency histogram.b Are the data symmetrical?c What is the modal class(es)?d What is the probability of more than

150 people attending a session?e What is the probability of having up to

100 people at a session?

8 Year 12 at Wallarwella High School sit exams in chemistry and mathematics. Theresults are shown in the table below.

a Is either distribution symmetrical? b State the mode of each distribution.c In which subject is the standard deviation greater? Explain your answer.

Mark Chemistry Maths

31–40 2 3

41–50 9 4

51–60 7 6

61–70 4 7

71–80 7 9

81–90 9 7

91–100 2 4

WORKEDExample

10No. of goals Frequency

11–20 3

21–30 6

31–40 7

41–50 23

51–60 21mmultiple choiceultiple choice

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1–50 2

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d The students were told that the probability of achieving more than 80% in chem-istry was the same as it was in mathematics. Is this true? Explain.

e Is the probability of obtaining more than a pass mark (50%) greater in chemistryor mathematics?

f What is the propability of achieving over 90% in each subject?

Note that some answers in the following questions may vary depending on the position ofthe line of best fit. The use of a graphics calculator is recommended, if available.

9 A drug company wishes to test the effectiveness of a drug to increase red blood cellcounts in people who have a low count. The following data were collected.

Construct a scatterplot, then draw in the line of best fit to find the red blood cell countat the beginning of the experiment (that is, on day 0).

10 A wildlife exhibition is held over 6 weekends and features still and live displays. Thenumber of live animals that are being exhibited varies each weekend. The number ofanimals participating, together with the number of visitors to the exhibition eachweekend, is shown in the table which follows.

Construct a scatterplot, then draw in the line of best fit to find the predicted number ofvisitors if there are no live animals.

11 A study of the dining-out habits of various income groups in a particular suburbproduces the results shown in the table below.

Use the data to predict:a the number of visits per year by a person on a weekly income of $680b the number of visits per year by a person on a weekly income of $2000.

12 The following table represents the costs for transporting a consignment of shoes fromBrisbane factories. The cost is given in terms of distance from Brisbane. There aretwo factories which can be used. The data are summarised below.

a Draw the line of best fit for each factory.b Which factory is likely to have the lowest cost to transport to a shop in Brisbane?c Which factory is likely to have the lowest cost to transport to Mytown, 115 kilo-

metres from Brisbane?d Which factory has the most ‘linear’ transport rates?

Day of experiment 4 5 6 7 8 9

Red blood cell count 210 240 230 260 260 290

Number of animals 6 4 8 5 7 6

Number of visitors 311 220 413 280 379 334

Weekly income ($) 100 200 300 400 500 600 700 800 900 1000

Number of restaurant visits per year 5.8 2.6 1.4 1.2 6 4.8 11.6 4.4 12.2 9

Distance from Brisbane (km) 10 20 30 40 50 60 70 80

Factory 1 cost ($) 70 70 90 100 110 120 150 180

Factory 2 cost ($) 70 75 80 100 100 115 125 135

WORKEDExample

11

Interpolation/Extrapolation

SkillSH

EET 4.4



Year 2000 Summer Olympic GamesIt has been claimed that the Summer Olympic Games, held in Sydney in the year 2000, were planned for September, because past records indicated that the chance of rain in Sydney during that month was low. (We recall the rain that fell at the time of one of Cathy Freeman’s 400 m races.)

Rainfall and temperature records for Sydney are available from the table below and graph at right.

The table shows the number of rainy days for Sydney for each month of the year for the years 1980 to 1990.

1 Identify the type of statistical calculation(s) that would be appropriate to perform on the data. Decide also on appropriate graphical representations.

2 Produce the graphs and perform your calculations.

3 Analyse your calculations and graphs. Comment on the claim made in the opening paragraph. Would an alternative month have been more appropriate, as far as the rainfall factor is concerned?

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– 20

– 10

0

10

20

30

40

50

Jan FebMarAprMayJun Jul AugSep Oct NovDec

Monthly temperatures Sydney

Month of yearHighest MaximumAverage Maximum

Lowest MinimumAverage Minimum

Tem

pera

ture

°cSydney rainfall (days of rain per month)

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1980 14 12 8 7 15 12 7 4 3 10 11 12

1981 15 15 5 8 14 7 6 7 6 12 17 13

1982 11 10 19 5 2 11 12 3 9 11 3 12

1983 9 12 9 12 13 8 8 12 9 16 8 18

1984 15 20 11 10 8 6 15 6 10 8 15 10

1985 9 10 8 19 13 6 7 10 12 18 19 16

1986 15 10 11 12 11 3 9 9 10 11 18 9

1987 14 11 10 7 9 11 11 16 4 17 8 16

1988 11 15 14 18 12 10 6 10 10 3 14 19

1989 21 11 22 23 22 20 12 9 7 6 13 15

1990 14 21 22 19 17 10 12 9 13 13 8 13



4 Does the month of September appear to be a suitable one for summer sporting events, when temperature is considered? Monthly temperature figures for Sydney are shown in the graph.

5 Describe the temperature conditions in Sydney during the month of September.

6 Compare rainfall and temperature figures for September with those for other months of the year.

7 Assume you are responsible for a submission to the Australian Olympic Committee recommending a suitable month in which to hold the summer Olympics. Prepare your report, remembering to support your recommendations by referring to the available data, calculations and graphs.

Sampling text to predict population characteristics

Research shows that the letter ‘e’ is the most frequently used letter in the English language. This activity aims to determine the frequency of its occurrence on a page of English text using a sample selected from the page.

1 Choose a book with an extended section of continuous prose. Select 20 full lines of text from a page (ignore incomplete lines).

2 Draw up the table below.

3 Count the number of e’s per line and complete the second column.

4 Using the figures in the second column, complete the ‘cumulated total’ column.

5 Use a graphics calculator, spreadsheet or graph paper to draw a scatterplot of the number of lines against the accumulated total.

6 Draw the line of best fit on your scatterplot.

7 Count the number of lines of text on your page. Extrapolate your scatterplot to estimate the number of e’s on the page you have chosen.

8 Compare your answer with those obtained from different texts by other members of your class. Comment on the similarities and variations in your answers. Factors that must be taken into account when making comparisons between the number of e’s per page of printed material include page size, font size, line length, line spacing and so on.

Aside: The novel, A void by George Perec (Harper Collins 1994) is written entirely without the letter ‘e’.

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Line number Number of e’s Cumulated total

123……20



Comparing population characteristicsThe following tables display data collated from the 1996 ABS Census relating to the populations in two Brisbane suburbs. Note: Income figures are weekly income, expressed in AUD.

Chapel HillTotal persons (a) 4 824 5 112 9 936Aged 15 years and over (a) 3 761 4 070 7 831Aboriginal 3 3 6Torres Strait Islander 0 0 0Both Aboriginal and Torres Strait Islander (b) 0 0 0Australian born 3 405 3 704 7 109Born overseas: Canada, Ireland, NZ, South Africa, UK (c) and USA 696 637 1 333Born overseas: Other country (d) 587 605 1 192Born overseas: Total 1 283 1 242 2 525Speaks English only and aged 5 years and over 3 936 4 230 8 166Speaks language other than English (e) and aged 5 years and over 459 491 950Australian citizen 4 239 4 515 8 754Australian citizen aged 18 years and over 3 027 3 315 6 342Unemployed 141 132 273Employed 2 677 2 427 5 104In the labour force 2 818 2 559 5 377Not in the labour force 854 1 393 2 247Enumerated in private dwelling (a) 4 815 5 085 9 900Enumerated in non-private dwelling (a) 9 27 36Persons enumerated same address 5 years ago 2 217 2 381 4 598Persons enumerated different address 5 years ago 2 163 2 324 4 487Overseas visitor 54 75 129

Chapel HillMedian age 34Median individual income 415Median household income 1 209Average household size 3.1

InalaTotal persons (a) 6 401 6 886 13 287Aged 15 years and over (a) 4 516 5 083 9 599Aboriginal 398 482 880Torres Strait Islander 42 51 93Both Aboriginal and Torres Strait Islander (b) 8 12 20Australian born 4 066 4 468 8 534Born overseas: Canada, Ireland, NZ, South Africa, UK (c) and USA 665 694 1 359Born overseas: Other country (d) 1 396 1 462 2 858Born overseas: Total 2 061 2 156 4 217Speaks English only and aged 5 years and over 4 000 4 483 8 483Speaks language other than English (e) and aged 5 years and over 1 454 1 494 2 948Australian citizen 5 424 5 807 11 231Australian citizen aged 18 years and over 3 521 4 001 7 522Unemployed 605 349 954Employed 2 036 1 403 3 439In the labour force 2 641 1 752 4 393Not in the labour force 1 728 3 144 4 872Enumerated in private dwelling (a) 6 397 6 886 13 283Enumerated in non-private dwelling (a) 4 0 4Persons enumerated same address 5 years ago 2 986 3 343 6 329Persons enumerated different address 5 years ago 2 319 2 493 4 812Overseas visitor 13 15 28

InalaMedian age 30Median individual income 187Median household income 412Average household size 2.8

1 Examine the data carefully.

2 Prepare a report comparing the residents of the two suburbs. Support your statements by reference to figures.

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Modelling Olympic Games timesThe running time for the men’s 100-m event in the Olympic Games broke the 10-second barrier for the first time in 1968 when Jim Hines of the United States clocked a time of 9.95 seconds. Knowing that records are broken over time, is it possible to predict a year when a runner could break the 9.5-second barrier?

We could model this situation by looking at past times for the event. These are shown in the table at right.Note: The times have not decreased consistently over the years. In fact, after 1968, the 10-second barrier was not broken again until 1984 when Carl Lewis of the United States won with a time of 9.99 s. For this reason, along with other factors which accompany feats of human endurance, in modelling situations such as this, any resulting predictions can only be considered as estimates.

1 Using a graphics calculator, spreadsheet or graph paper, draw a scatterplot of ‘year’ (on the x-axis) against ‘time’ (on the y-axis). This could be viewed as a time series as the years are at equal intervals.

2 Draw the line of best fit for your scatterplot.

3 Use your line of best fit to find the year in which the 9.5-second barrier could be broken in the 100-m sprint. This represents a situation when we are extrapolating data (that is, determining a value outside the range of data plotted). Note that the Olympic Games are only held every four years and that your answer may not be one of those years. It remains to be seen how accurate your prediction is!

4 Use some resources available to you (World Wide Web, reference books, almanacs and so on) to collect data on other sporting events (such as high jump heights, shot put distances and swimming times). Model the situation, enabling a prediction to be proposed.

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Year Time (seconds)

1948 10.30

1952 10.40

1956 10.50

1960 10.20

1964 10.00

1968 9.95

1972 10.14

1976 10.06

1980 10.23

1984 9.99

1988 9.92

1992 9.96

1996 9.84

2000 9.87



Predicting test resultsOver the year Sally had sat for nine mathematics tests, but had been sick at the time of the tenth test. She had achieved above average marks for each of her nine tests, so her teacher did not want to give her the class average for her last test. As sometimes happens, the class as a whole found this last test more difficult than the previous ones, so generally all marks were depressed. It would not then be fair to give Sally the average of her previous tests for this last one. How could her teacher give Sally an estimated mark based on her past performance compared with that of her fellow students?

Below is a table displaying the class average percentage for each test and Sally’s percentage on the tests.

Because Sally’s marks were consistently above the class average by 10% or more, we could explore a relationship between the class average and Sally’s marks.1 Using a graphics calculator, or otherwise, construct a scatterplot of the first nine

tests, plotting the class average percentage on the x-axis and Sally’s percentage on the y-axis.

2 Draw in the line of best fit.3 Use your line of best fit and a value of 49 for the class average percentage to

determine an estimate for Sally’s performance had she sat for the tenth test.4 Do you think this is a fair mark to award to Sally? Justify your answer.This method can be used in a variety of situations to estimate values for missing results. The predictions become less accurate if there is a great deal of inconsistency in past performance.

The door gameImagine you are a contestant in a game on television. The conditions and rules are as follows:1. Three doors (door 1, door 2, and door 3) stand before you.2. Behind one of these doors which face you and the audience lies the prize of

your dreams (a trip to Wimbledon or a Ferrari car). (The organisers know where the prize lies.)

3. Behind each of the other two doors there is nothing.4. You see the three closed doors before you and you have no hints.5. With the benefit (or distraction) of audience participation, you are asked to

select the door behind which you think the prize lies.6. You tell the compere and audience which door you choose.7. Before opening the door you have chosen, the compere tells you that first, one

of the doors, different from the one you have chosen, will be opened; it will be a door that does not have the prize behind it.

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Test 1 2 3 4 5 6 7 8 9 10

Class % 64 59 60 55 62 66 58 63 65 49

Sally’s % 72 70 77 65 75 80 71 75 75

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8. There are two doors now unopened and you are given an offer to change your choice.

9. Of the two unopened doors, one is your original choice and the other not your choice.

10. Using the theory of probability, would you have a better chance of winning if you stayed with your original choice, or made the switch?

This was a game which was actually played on television for a lengthy period many years ago. Regular viewers of the program formed their own opinions regarding the better option by analysing the results of contestants’ choices. The background to the game can be researched by a web search of the name Monty Hall. You may be interested to conduct a search at this stage, or leave your search until you have had time to formulate a logical reason for your choice.

Part IThis activity provides practical experience for the door game. The simulation could be undertaken as a class activity, or in pairs.

1 Simulate three doors using books lying on a flat surface instead of doors.

2 Underneath one of them place a small piece of paper. (The contestant is to be unaware of the location of the paper.)

3 Ask the contestant to select one of the three books underneath which he or she thinks this piece of paper lies.

4 Turn over one book other than one chosen by the contestant and underneath which the object does not lie.

5 Ask the contestant whether he/she would like to choose a different book.

6 Turn over the book under which the paper lies.

7 Repeat the simulation ten times. Copy and complete the table below, recording your results after each turn shown in the table.

Game Stay with choice Change mind Win/Lose

1 ✔ ✔

2 ✔ ✖

3 ✔ ✔

4

5

6

7

8

9

10



8 Combine your results with those of other members of your class so that your combined set consists of a large number of simulations.

9 What are your conclusions from your experiment? Is it wise to stay with your original choice or should you change your mind?

If you conduct a web search, you will find sites which simulate the door game using an interactive mode.

One such site can be found at www.jaconline.com.au/maths/weblinks.

Part IILet us consider the probabilities of the choices in the ‘door game’.

1. Since there are three doors, the probability of the prize being behind any of these doors is . This enables us to start a tree diagram as shown at right.

2. As you have three doors from which to select, the probability that your choice will be the correct selection is . This realisation enables us to extend the branches of the tree as at right.

3. Multiplying the probabilities along the branches enables us to fill in the probability for each selection (shown at right).

1

2

3

1–3

1–3

1–3

Prize door13---

1123

2

3

1–3

1–31–3

1–3

1–3

1–31–3

1–3

1–3

Prize door

Your choice 123

123

1–3

1–3

1–3

13---

1

123

123

123

2

3

1–3

1–9

1–9

1–9

1–9

1–9

1–9

1–9

1–9

1–9

1–3

1–3

Prize door

Your choice




For the set of scores 23, 45, 24, 19, 22, 16, 16, 27, 20, 21, use your calculator to find:

1 the mean 2 the median

3 the mode 4 the range

5 the interquartile range 6 the standard deviation.

7 In the data set, is the distribution symmetrical?

8 Does the data set have an outlier (a score which is not typical of the data set)?

9 Which measure of central tendency is the best measure of location in this data set?

10 Explain why the interquartile range is a better measure of spread than the range.

4. The next stage is the crucial factor. Depending on which door is opened, will you win if you stay with your choice, or are you more likely to win if you change your mind? Follow the results completed for door 1. The combinations of opened doors have been completed below for the remainder of the tree. Complete the blank spaces.

Prize door Your choice Door opened Win if Win if I I stay? change my mind?

2 or 3 Yes (P = ) No

3 No Yes (P = )

2 No Yes (P = )

3 ____ ____

1 or 3 ____ ____

1 ____ ____

2 ____ ____

1 ____ ____

1 or 2 ____ ____

5. Add the probabilities in the ‘Win if I stay?’ column. This gives the overall probability of your winning if you stay with your original choice.

6. Similarly, add the probabilities in the ‘Win if I change mind?’ column. This figure represents the overall probability of your winning if you change your mind and choose the other unopened door.

1 Based on these probabilities, which choice should you make? How does this agree with your experimental results from your practical simulation?

If you have not yet visited websites by searching for Monty Hall, it would now be appropriate to do so. This game has created much discussion among great mathematicians and non-mathematicians over many years.

1

123

123

123

2

3

1–3

1–9

1–9

1–9

1–9

1–9

1–9

1–9

1–9

1–9

1–3

1–3

19---

19---

19---

WorkS

HEET 4.3

2



Data collection• A statistical investigation can be done by either census or sample.• A census is when an entire population takes part in the investigation.• A sample is when a small group takes part in the investigation and the results are

taken to be representative of the whole group.• There are many types of sample.

1. Random sample — chance is the only factor in deciding who participates.2. Stratified sample — the sample taken is chosen so that it has the same

characteristics as the whole population.3. Systematic sample — there is a method for deciding who participates in the

sample.4. Accessibility sample — those easily accessed form the sample.5. Quota sample — the number in the sample is limited by the quota.6. Judgemental sample — a judgement is made when deciding the sample.7. Cluster sample — clusters within the population form the sample.

BiasIf the sample is poorly chosen the results of the investigation will be biased. This means that the results will be skewed towards one section of the population.

Estimating populationsPopulations that can’t be accurately counted with ease are estimated by using the capture–recapture technique.

Contingency tables• Display horizontal and vertical categories of two variables of a set of data• Calculations with regard to a variety of reference bases can be made.

Applications of statistics and probability• Frequency histograms can be used to estimate probabilities in data sets.• Scatterplots display the relationship between two variables and allow predictions

to be made.

Measures of central tendency and spread• The mean, median and mode are measures of central tendency in a data set.• The mean is calculated by adding the scores then dividing by the number of scores.• The median is the middle score or average of two middle scores.• The mode is the most frequently occurring score.• The range, interquartile range and standard deviation are measures of spread.• The range is the difference between the highest and lowest scores.• The interquartile range is the difference between the upper and lower quartiles.• The standard deviation is found using the population or sample functions of a

calculator.• An outlier is a score that is much less or much greater than the other scores.

summary



1 For each of the following statistical investigations, state whether a census or a survey has been used.a The average price of petrol in Townsville was estimated by averaging the price at 40

petrol stations.b The Australian Bureau of Statistics has every household in Australia complete an

information form once every five years.c The performance of a cricketer is measured by looking at his performance in every

match he has played.d Public opinion on an issue is sought by a telephone poll of 2000 homes.

2

Which of the following is an example of a census?A A newspaper conducts an opinion poll of 2000 people.B A product survey of 1000 homes to determine what brand of washing powder is usedC Every 200th jar of Vegemite is tested to see if it is the correct mass.D A federal election

3 Name and describe 5 different methods for selecting a sample.

4 Which method of sampling has been used for each of the following?a The quality-control department of a tyre manufacturing company road tests every 50th

tyre that comes off the production line.b To select the students to participate in a survey, a spreadsheet random number generator

selects the roll numbers of 50 students.c An equal number of men and women are chosen to participate in a survey on fashion.

5 Use your random number generator to select 10 numbers between 1 and 1000.

6 The table at right shows the number of students in each year of school. In a survey of the school population, how many students from each year should be chosen, if a sample of 60 is selected using a stratified sample?

7 To estimate the fish population of a lake, 100 fish are caught, tagged and released. The next day another 100 are caught and it is noted that 5 have tags. Estimate the population of the lake.

4A

CHAPTERreview

4A mmultiple choiceultiple choice

4B4B

4B4B Year No. of students

7 212

8 200

9 189

10 175

11 133

12 124

4B


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 2278 Kimberley has a worm farm. To estimate the population of her farm, she catches 150 worms

and tags them before releasing them. The next day, she catches 120 worms and finds that 24 of them have tags. Estimate the population of the worm farm.

9 A sample of 200 fish are caught, tagged and released back into the population. Later Barry, Viet and Mustafa each catch a sample of fish.Barry caught 40 fish and 3 had tags.Viet caught 75 fish and 9 had tags.Mustafa caught 55 fish and 7 had tags.a Find the estimate of the population that each would have calculated.b Give an estimate for the population, based on all three samples.

10

Which of the following is an example of a random sample?A The first 50 students to arrive at school take a survey.B Fifty students’ names are drawn from a hat and those drawn take the survey.C Ten students from each year of the school are asked to complete a survey.D One class in the school is asked to complete the survey.

11 Carolyn is a marine biologist. She spends the day on a boat and 500 fish are netted. Carolyn notes the types of fish netted. There are 173 blackfish, 219 drummer and 108 mullet. The fish are tagged and released back into the school from which they were caught. Another 250 are then caught and it is noted that 63 have tags. Estimate the population of the school.

12 Bias can be introduced into statistics through:a questionnaire designb sample selectionc interpretation of statistical results.Discuss how bias could be a result of techniques in the above three areas.

13 A medical test screens 200 people for a virus. A positive test result indicates that the patient has the virus.1. Of 50 people known to have the virus, the test produced 48 positive results.2. Of the remainder who were known not to have the virus, the test produced one positive

result.Use the above information to complete the table below.

14 The results of a lie detector test are given below.1. Of 80 people who are known to be telling the truth, the lie detector indicates that three

are lying.2. Of 20 people known to be lying, the lie detector indicates that 17 are lying.Display this information in a contingency table.

Test results


With virus

Without virus

Total

4B

4B

4Bmmultiple choiceultiple choice

4B

4C

4D

4D



15 Below are the results of a test screening for a disease. A positive test indicates that the patient has the disease.

a How many people were tested for the disease?b How many positive test results were recorded?c What percentage of those people with the disease were correctly diagnosed by

the test?d If a person without the disease is chosen at random, what percentage returned a positive

test?

16 A reading test for people with dyslexia is given and the results are shown in the contingency table below.

a How many people were tested?b What percentage of people tested positive to dyslexia?c Based on the above results, if a person with dyslexia takes the test, what is the

percentage chance that they will be accurately diagnosed?

17

The contingency table below shows the results of a trial on new metal detectors for aircraft. The metal detector scans a piece of hand luggage and lights up if metal is found.

Based on the above results, the chance of metal going undetected in a piece of hand luggage is:A 10% B 25% C 75% D 90%

Test results




Total 126 14

Test results


With dyslexia 39 1 40

Without dyslexia 85 5 90

Total 124 6

Test results


With metal 9 1 10

Without metal 87 3 90

Total 96 4

4D

4D

4D mmultiple choiceultiple choice


C h a p t e r 4 P o p u l a t i o n s , s a m p l e s , s t a t i s t i c s a n d p r o b a b i l i t y 22918 A medical test for a disease does not always give the correct result. A positive test indicates

that the patient has the disease. The contingency table below shows the results of a new screening test for the disease. It was tested on a group of people, some of whom were known to be suffering from the disease, some of whom were not.

a How many people were tested for the disease?b What percentage of the results were accurate?c How many patients tested positive to the disease?d What percentage of patients with the disease were correctly diagnosed by the new test?e Based on the above results, what is the probability that a patient with the disease will

have the disease detected by this test?

19 The contingency table below compares the number of men and women who are right- and left-handed.

a What percentage of males are left-handed?b What percentage of females are left-handed?c Based on the above data, is there any significant difference between the percentage of

male and female left-handers?

20

The contingency table below shows the number of men and women who work in excess of 45 hours per week.

The percentage of men who work greater than 45 hours per week is closest to:A 28% B 34% C 51% D 67%

Test results




Total 192 8

Men Women Totals

Right-handed 158 172 330

Left-handed 17 15 32

Totals 175 187 362

Men Women Totals

≤45 hours 132 128 260

>45 hours 69 34 103

Totals 201 162 363

4D

4D

4Dmmultiple choiceultiple choice



21 Sixty-seven primary and 47 secondary school students were asked their attitude to the number of school holidays which should be given. They were asked whether there should be more, fewer or the same number. Five primary students and 2 secondary students wanted fewer holidays, 29 primary and 9 secondary students thought that they had enough holidays (that is, they chose the same number) and the rest thought that they needed to be given more holidays.a Form a contingency table with reference to

primary and secondary school percentages.b Use your table to compare the opinions of

primary and secondary school students.

22 Consider the data set represented by the frequency histogram on the right.a Are the data symmetrical?b Can the mean and median of the data be seen?c What is the mode of the data?d Which score has the highest probability of occurring?

23 The table below shows the number of attempts that 20 members of a Year 12 class took to obtain a driver’s licence.

a Show these data in a frequency histogram.b Are the data symmetrical?c What is the probability of a student of the class taking more than three attempts to obtain

a driver’s licence?

24 Consider this data set which measures the sales figures for a new salesperson.

a Draw the line of best fit.b Use your line to predict the sales figures for the tenth day.

Number of attempts Frequency

1 11

2 4

3 2

4 2

5 0

6 1

Day (d) 1 2 3 4 5 6 7 8

Units sold (s) 1 2 4 9 20 44 84 124

4D

4E

15 16

Freq

uenc

y

1718 19 2001234

7

56

8

Score

4E

testtest

CHAPTERyyourselfourself

testyyourselfourself

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