MTH3003 TUTORIAL QUESTIONS

FOR

UPM STUDENTS JULY 2010/2011

(PLEASE TRY ALL YOUR BEST TO ANSWER ALL QUESTIONS)

Preparation tutorial notes for Statistics of Applied Sciencesby Dr. Mohd Bakri Adamfor MTH3003 students

at Mathematics Department, UPM,Semester July 2010/2011.

Copyright c© 2010 by Mohd Bakri Adam

Contents

0.1 Tutorial 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Tutorial 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Tutorial 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4 Tutorial 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.5 Tutorial 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90.6 Tutorial 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.7 Tutorial 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140.8 Tutorial 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160.9 Tutorial 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190.10 Tutorial 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200.11 Binomial Table and Standard Normal Table . . . . . . . . . . . . . . . . . 24

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0.1 Tutorial 1

1. In a recent survey, 1000 adults in Malaysia were asked if they read news on theinternet at least once a week. Six hundred of the adults said yes. Identify thepopulation and the sample. Describe the data set.

2. Decide wether the numerical value describes a population parameter or a samplestatistic. Explain your reasoning.

(a) A survey of a sample of Master of Applied Statistics reported that the averagestarting salary for Master of Applied Statistics is less than RM3000.00.

(b) Starting salaries for the 667 Master of Applied Statistics graduates from UPMincreased 8.5% from the previous year.

(c) In a random check of a sample of retail stores, the health and consumer minis-tery found that 34% of the stores were not storing fish a the proper temperature.

3. What is a random variable? Give an example of discrete random variable and acontinuous random variable. Justify your answer.

4. Decide which part of the study represent the descriptive branch of statistics. Whatconclusions might be drawn from the study using inferential statistics?

(a) A large sample of women, aged 50, was studied for 18 years. For unmarriedwomen, approximately 70% were alive at the age 65. For married women, 90%were alive at the age 65.

(b) In a sample of BSKL analysts, the percentage who incorrectly forecasted high-tech earnings in a recent year was 45%.

0.2 Tutorial 2

Table 1: The number of minutes 50 students spent on reading during their leasure time

50 40 41 17 11 7 22 44 28 2119 23 37 51 54 42 88 41 78 5672 56 17 7 69 30 80 56 29 3346 31 39 20 18 29 34 59 73 7736 39 30 62 54 67 39 31 53 44

1. From Table 1 construct a frequency distribution using Sturges method. Where itused the following formula to estimate the number of the classes,

Class Numbers, k = 1 + 3.322 log10(n)

1

where n is the sample size and k is the round up value from the formula. Use thesmallest value in the data as a lower class limit.

2. Construct a frequency distribution that has five classes from Table 1, include addi-tional features such as midpoint, relative frequency, cumulative frequency, cumula-tive relative frequency.

3. From question 1 construct a histogram, polygon and ogive.

4. Calculate mean, mode and median of the data from Table 1.

5. Calculate mean, mode and median from frequency table in question 2. Hint: Useformula in Question 7. It will be tested in the examination.

6. Calculate P30, P45, P80 for the data in Table 1.

Class Frequency, Classf Boundaries

7-18 6 6.5-18.519-30 10 18.5-30.531-42 13 30.5-42.543-54 8 42.5-54.555-66 5 54.5-66.567-78 6 66.5-78.579-90 2 78.5-90.5

Table 2: Frequency distribution for reading times (in minutes) with boundaries.

7. Calculate P30, P45, P80 for the data in frequency table in Table 2. Used the followingformula

Pp = LCBp +[n(0.p)− CFBPC]w

fp

where

LCBp is the lower class boundary which equavalance to the cumulative frequencyclass equal to 0.p or the nearest value greater than 0.p.

CFBPC is the cumulative frequency before percentile class.

w is width of the percentile class.

fp is a frequency in the percentile class.

As a Pp for a grouped calculation is not given in the notes, you need to go to thelibrary of UPM to look for the references. It will be tested in the examination.

8. Calculate a variance for Table 3. As a variance for a grouped calculation is notgiven in the notes, you need to go to the library of UPM to look for the references.It will be tested in the examination.

2

Table 3: Frequency distribution for reading times (in minutes) with midpoint, rf and cf .

Class Frequency, Midpoint Relative Cumulativef frequency frequency

7-18 6 12.5 0.12 619-30 10 24.5 0.20 1631-42 13 36.5 0.26 2943-54 8 48.5 0.16 3755-66 5 60.5 0.10 4267-78 6 72.5 0.12 4879-90 2 84.5 0.04 50∑

f = 50∑f/n = 1

9. Construct stem-and-leaf for the data from Table 1.

10. Construct box-and-whiskers plot for the data from Table 1

11. Suppose the age of a sample of 10 students are:

20.9, 18.1, 18.5, 21.3, 19.4, 25.3, 22.0, 23.1, 23.9, and 22.5

(a) Calculate the mean, mode and median.

(b) Calculate the range, variance, coefficient of variation and standard deviation.

(c) Find x1, x3 and x(3) values.

(d) Construct a stem-and-leaf.

(e) Calculate P40, D1 and D7.

(f) Construct the box-and-whiskers plot. What is you comment on the plot.

12. What is the statistic(s) that you use in the following scenario

(a) You are working in the supermarket and you want to advise you manager inpurchasing a new stocks of shampoo.

(b) You want to conclude about a performance of MTH3003 in your class.

(c) You were asked by your neigbour about planting manggo tree in your backyard.

(d) You were asked to choose one bag from 4 different bags.

(e) You’re now a lecturer for MTH3003, the top management wants to imposednew system of grading i.e. Gred A is given only to ten percents of all studentsfrom the highest mark and gred F is given only to the last five percent ofstudents from the lowest marks.

(f) You want to by a new bed and a new furniture for your new rental room in SriSerdang.

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(g) You have asked your senior regarding his study times at University.

(h) You want to report the everage times of students using the library facilities ina year.

13. The following unordered stem-and-leaf display represents the number of hours spentby 30 MTH3003 students working on assignments during the past month.(1|0 rep-resents 1 units)

1 | 0 72 | 6 2 03 | 2 7 4 84 | 5 1 0 2 9 65 | 9 9 1 2 3 6 6 4 86 | 5 4 4 27 | 8 5

(a) Construct a frequency table and histogram for the about table with a numberof classes is 7.

(b) Construct an Ogive.

(c) Calculate the mean, mode, variance and coefficient of variation for the groupeddata in (a). How is the dispersion of the data?

(d) Calculate the median for the grouped data.

14. For a given data as follows

0 4 3 2 8 9 7 6 6 0

(a) Calculate the sample mean, mode and median

(b) Calculate the sample variance, standard deviation and the coefficient of varia-tion.

(c) Construct a box-and-whiskers plot. What you comment on the skewedness ofthe data.

0.3 Tutorial 3

1. How many ways to arrange 6 person in a round table.

2. How many ways to arrange 2 pairs of twin in a round table.

3. Identify the sample space of the probability experiment

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(a) Tossing a coin.

(b) Answering a true and false question.

(c) Tossing four coins and recording the number of heads.

(d) Answering a multiple choice question with A, B, C, and D as the possibleanswers.

(e) Determining the children’s gender for a family of three children.

(f) Rolling a single 12-sided die with sides numbered 1-12.

(g) A calculator has a function button to generate a random integer from -5 to 5.

4. A coin is tossed. Find the probability that the result is heads. Answer:0.5

5. A single six-sided die is rolled. Find the probability of rolling an even number.Answer:0.5

6. If an individual is selected at random, what is the probability that he or she has abirthday in July? Ignore leap years. Answer: 31

365

7. A question has five multiple-choice questions. Find the probability of guessing thecorrect answer. Answer:0.2

8. Find the probability of selecting two consecutive threes when two cards are drawnwithout replacement from a standard deck of 52 playing cards. Round your answerto four decimal places. Answer: 0.0045

9. A multiple-choice test has five questions, each with five choices for the answer. Onlyone of the choices is correct. You randomly guess the answer to each question.

(a) What is the probability that you answer the first two questions correctly?Answer:0.04

(b) What is the probability that you answer all five questions correctly? An-swer:0.00032

(c) What is the probability that you do not answer any of the questions correctly?Answer:0.32768

(d) What is the probability that you answer at least one of the questions correctly?Answer:0.67232

10. The events A and B are mutually exclusive. If Pr(A) = 0.3 and Pr(B) = 0.2, whatis Pr(A ∪B). Answer:0.5

11. Given that Pr(A ∪B) = 13,Pr(A) = 1

4, and Pr(A ∩B) = 1

5, find Pr(B). Answer:17

60

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12. A tourist in Malaysia wants to visit six different cities i.e. Kuala Lumpur, Penang,Shah Alam, Johor Bharu, Seremban and Kota Bharu.

(a) How many different routes are possible? Answer:720

(b) If the route is randomly selected, what is the probability that the tourist willvisit the cities in alphabetical order? Answer: 0.001

13. Four best friends of MTH3003 are invited for a dinner by their lecturer. How manyways can they be seated at a dinner table if the table is straight with seats only onone side. Answer:24

14. Yesterday five best friends of MTH3003 have been invited for a dinner by theirlecturer. How many ways can they be seated at a dinner table if the table is straightwith seats on only one side. Answer:120

15. How many ways can jury of six men and six women be selected from twelve menand ten women? Answer: 194,040

16. How many different permutations of the letters in the word PROBABILITY arethere? Answer: 9,979,200

17. The acces code to a house’s security system consist of five digits.

(a) How many different codes are available if each digit can be repeated? An-swer:100,000

(b) How many different codes are availbe if the first digit cannot be zero and thearrangement of five fives is excluded? Answer:89,999

18. In Malaysia, each automobile license plate consists of a single digit followed by threeletters, followed by three digits.

(a) How many distinct license plates can be formed if there are no restrictions onthe digits or letters? Answer:175,760,000

(b) How many distinct license plates can be formed if the first number cannot bezero and the three letters cannot form "GOD"? Answer: 158,175,000

19. An experiment involves tossing two dice. Find the following events

(a) List the sample space.

(b) The probability that sum is at least 3.

(c) The probability that both dice show odd numbers.

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20. Urn A contains three red balls and seven blue balls. Urn B contains eight red ballsand four blue balls. Urn C contains five red balls and eleven blue balls. An urn ischosen, with each urn equally likely to be chosen, and then a ball is chosen randomlyfrom that selected urn. Calculate the probabilities

(a) A red ball is chosen.

(b) A blue ball is chosen.

(c) A red ball from urn B is chosen.

0.4 Tutorial 4

1. Three objects were selected by random from a container. Each object taken waschecked and categorised either good or bad. If X represents the good one, what isthe probable values of X?

2. State whether the variable is discrete of continuous

(a) The number of cups of coffee sold in a cafeteria during lunch.

(b) The height of a player on a basketball team.

(c) The cost of a statistics book.

(d) The blood pressures of a group of students the day before their final exam.

(e) The temperature in degrees Fahrenheit on July 4th in Kuala Lumpur.

(f) The number of goal scored in a soccer game.

(g) The speed of a car on a Seremban-KL highway during rush hour traffic.

(h) The number of phone calls to the attendance office of a high school on anygiven school day.

(i) The age of the oldest student in MTH3003 class.

3. Determine whether the distribution represents a probability distribution. If not,identify the requirement that is not satisfied.

(a)x 1 2 3 4 5

Pr(x) 0.2 0.2 0.2 0.2 0.2

(b)x 3 6 9 12 15

Pr(x) -0.3 0.5 0.1 0.3 0.4

(c)x 1 2 3 4 5

Pr(x) 1.2 1.2 1.4 1.1 1.1

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4. Construct a distribution for the formula and determine whether it is a probabilitydistribution.

(a) Pr(x) = x6, x = 1, 2, 3.

(b) Pr(x) = x6, x = 3, 4, 7.

(c) Pr(x) = xx+2

, x = 0, 1, 2.

5. The random variable X represents the number of boys in a family of three children.Assuming that boys and girls are equally likely.

(a) Construct a probability distribution.

(b) Graph the distribution.

(c) Find the mean and standard deviation for the random variable X.Answers: 1.50,0.87

6. A twenty-five-years-old man decides to pay RM325 for a one-year insurance policywith coverage for RM1,000,000. The probability of him living through the year is0.9995. What is his expected value for the insurance policy? Answer:RM175.16

7. One thousand tickets are sold at RM1 each. One ticket will be randomly selectedand the winner will receive a color television valued at RM350. What is the expectedvalue if a person buys one tickets? Answer:-RM0.65

8. If a person rolls doubles when tossing two dice, the roller profits RM5. If the gameis fair, how much should the person pay to play the game? Answer:RM1

9. Show that f(x) is pdf.

f(x) =

x2, for 0 < x < 2

0, otherwise .

10. Find the k value if

f(x) =

kx2 , for 0 < x < 4

0, otherwise .

is a density probability function.

11. As f(x) is pdf as follow

f(x) =

32

√x, for 0 < x < 1

0, otherwise .

(a) Find F (x) function

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(b) Find Pr(0.2 < x < 0.3)

(c) Find F (0.3)

(d) Find Pr(x > 0.3)

(e) Find the expected value.

(f) Find the variance, V ar(x).

12. Find the k value if

f(x) =

(x2 + 3)k, for − 2 < x < 3

0, otherwise .

is a density probability function. Then answers the following questions

(a) Find Pr(1 < x ≤ 2)

(b) Find Pr(x > 2)

(c) Find the expected value.

(d) Find the variance.

0.5 Tutorial 5

1. Decide whether the experiment is a binomial experiment. If it is not, explain why.

(a) You observe the gender of the next 100 babies born at a local hospital. Therandom variable represents the number of girls.

(b) You roll a die 100 times. The random variable represents the number thatappears on each roll of the die.

(c) You spin a number wheel that has 10 numbers 100 times. The random variablerepresents the winning numbers on each spin of the wheel.

(d) Surveying 100 prisoner to see whether they are serving time for their firstoffense. The random variable represents the number of prisoners serving timefor their first offense.

(e) Selecting five cards, one at a time without replacement from a standard deckof cards. The random variable is the number of red cards.

2. Assume that male and female births are equally likely and that the birth of anychild does not affect the probability of the gender of any other children. Find theprobability of at most three boys in ten births. Answer:0.172

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3. A test consist of 10 true or false questions. To pass the test a student must answerat least eight questions correctly. If the student guesses on each question, what isthe probability that the student will pass the test? Answer:0.055

4. A test of consist of 10 true of false questions. If the student guesses on each question,what is the mean and standard deviation of the number of correct answers?5;1.58

5. A company ships computer components in boxes that contain 20 items. Assumethat the probability of a defective computer component is 0.2. Find the probabilitythat the first defect is found in the seventh component tested. Round your answerto four decimal places. Answer:0.0524

6. A sales firm recieves an average of three calls per hour on its toll-free number. Forany given hour, find the probability that it will receive exactly three calls and findthe probability that it will receive at least three calls. Answers:0.2240;0.5768

7. A local fire station recieves an average of 0.55 rescue calls per day. Find the prob-ability that on a randomly selected day, the fire station will receive fewer than twocalls. Answer:0.894

8. Given that a marksman can hit a target on a single trial with probability equal to0.8. Suppose he fires 4 shots at the target. What is the probability that he will hitthe target at least once. Binomial,0.1536

9. The mean number of bacteria per ml of liquid is 2. By assuming that the numberof bacteria follows a Poisson distribution, find the probability that in 1 ml of liquidthere are 3 bacteria. 0.1804

10. A factory manufactures computer and 100 computers are packed in a box. Theprobability that a computer is defective is 0.005. Find the probability that thereare 2 defective computers in a box. Approximation Poisson, 0.0758

11. The time taken by Aminah to distribute Nasi Lemak everyday to residents stayingat Fifth College follows a normal distribution with mean 32 minutes and variance6.25 minutes2. Find the probability that on Wednesday, Aminah takes more than40 minutes.Normal,0.000687

12. A regular tetrahedral shaped die with its fares labelled 1,2,3 and 4 is tossed 200times. Find the probability of obtaining more than 60 times the digit 4. Approxi-mation Normal,0.0432

13. The waiting time at a clinic MTH3003 has an exponential distribution with anaverage waiting time of 5 minutes. What is the probability that you have to wait10 minutes or more at the counter. 0.135

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14. When a blue bulb is selected at random from a box that contains a 40 watt bulb,a 60 watt bulb, a 75 watt bulb, and a 100 watt bulb. Each elements of the samplespace S = {40, 60, 75, 100} occurs ith the probability 1/4. Derived the distributioninvolved.

15. The complexity of arrivals and departures into an airport are such that computersimulation is often used to model the ideal condition. For a certain airport containingthree runways it is known that in the ideal setting the following are the probabilitiesthat the individual runaways are access by a randomly arriving commercial aircraft.

• Runaway 1: P1 = 2/9

• Runaway 2: P2 = 1/6

• Runaway 3: P3 = 11/18

What is the probability that 6 randomly arriving aircrafts are distributed in thefollowing pattern.

• Runaway 1: 2 aircarfts

• Runaway 2: 1 aircraft

• Runaway 3: 3 aircraft

Multinomial,0.1127

16. We have 100 items of which 12 are defectives. What is the probability that in asample of 10, 3 are defectives? Hypergeometry,0.08

17. In MTH3003 soccer league series, the team who wins 4 games out of 7 will be thewinner. Suppose the team A has probability 0.55 of winning over team B and bothteams A and B faced each other in the league series. What is the probability thatteam A will win the league series in six games? Negative binomial,0.1853

18. In a certain manufacturing process it is know that, on the average, one in every 100items is defective. What is the probability that the fifth item inspected is the firstdefective item. Geometric,0.0096

19. During the laboratory experiment the average number of radioactive particle passingthrough a counter in one millisecond is 4. What is the probability that 6 particlesenter the counter in a given millisecond. Poisson,0.1042

20. Suppose we select 5 cards from an ordinary deck of playing cards. What is theprobability of obtaining 2 or fewer hearts? Hypergeometry, 0.9072

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21. (a) A test bank consists of 1000 objective questions. Any student may randomlyselect any 10 questions to work on. Each question has 5 choices (A, B, C, D,E) and there is exactly one correct answer to a question. Let X be the numberof correctly answered questions for a particular student. Calculate Pr(X > 4).

(b) A bowl contains 5 blues balls and 6 red balls of the same size. Four balls areselected at random from the bowl. Let Y be the number of blue balls in thesample selected. Find Pr(Y ≤ 3) and calculate E(Y ) and V ar(Y ).

(c) An external examiner realized that it is usual to observe, on the average, fivetyping errors on every page of a thesis submitted for the first time. if Xrepresents the variable mentioned, calculate Pr(3 < X ≤ 6).

22. (a) The weights of newly born baby girls in Hospital Serdang may be assumed tobe normally distributed with mean 3.5 kg and standard deviation of 1.2 kg. Ifa baby girl born at that hospital is randomly selected, what is the probabilitythat its weight is within the range from 2.0 to 4.0 kg?

23. Suppose that a baby girl born at the hospital stated in the previous question.

(a) will be incubated if its weight does not exceed 2.5 kg. Previous records in-dicated that 3 out of 10 baby girls born at that hospital needed incubation.If the records of 100 baby girls born at that hospital are selected at random,approximate the probability that the number of baby girls who had undergoneincubation is between 24 and 32.

(b) may have congenitials defects. Previous records indicated that only 1% of allbaby girls born at that hospital had congenital defects. One hundred recordsof babies born at that hospital are selected at random, approximate the prob-ability that more than 2 had congenital defects.

0.6 Tutorial 6

1. The weight of the packages acceptance by one big hypermarket is normally dis-tributed with mean 30 kg and the standard deviation is 5 kg. What is the proba-bility that 25 packages which accepted randomly will exceed the maximum limit of820 kg, for the lift carrier in the building of the hypermarket? Answer:0.0026

2. In one of random sample of size 25 from normally distributed with mean 98.6 andstandard devation 17.2. Answers the following questions.

(a) Pr(92 < X < 102)

(b) Find the probability of the previous question if n now is 36.

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3. Given X ∼ N(5, 36)

(a) Find E(X) and V (X) if n = 20.

(b) Find Pr(X > 7) if n = 6.

4. The mean for an executive paying for his lunch is RM 6.50 with standard deviationis RM 6.00. If 36 executives are selected at random from the firm (i.e. where theexecutive working), find the probability that the mean money spent are betweenRM 5.00 and RM 10.00.

5. Mr Ali, is an auditor in a credit card company, knows that in average a client havethe balance in the account is RM 112 and standard deviation is RM 56. If he audited50 accounts which he selected by random, what is the probability that the averagebalance in a month is below RM 100. Answer:0.0643

6. The height for 1000 trees is approximating normal distribution with mean 174.5 cmand standard deviation is 6.9 cm. If 200 sample which each is n = 25 where takenfrom the population and the mean where recorded

(a) Find the mean and standard deviation for X.

(b) The sample size, when mean is between 172.5 and 175.5 cm. Answer:151 trees

(c) The sample size if th emean is below 172.0. Answer:8 trees

7. The balances of all current accounts at a local bank belong to one of former MTH3003have a distribution that is skewed to the right with its mean equal to RM12,450 andstandard deviation equal to RM4,300. Find the probability that the mean balanceof a sample of 50 current accounts selected from this bank will be

(a) more than RM11,500.

(b) between RM12,000 and RM13,800.

(c) within RM1,500 of the population mean.

(d) more than the population mean by at least RM1,000.

8. Kuala Lumput city council is planning to build a gigantic solar-power plant. A localnewspaper found out that 53% of the voters in the city favour the construction ofthis plant. Assume that this result holds true for the population of all voters in thiscity.

(a) What is the probability that more than 50% of the voters in a random sampleof 200 voters selected from this city will favour the construction of this plant?

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(b) A politician would like to take a random sample of voters in which more than50% would favour the plant construction. How large a sample would be selectedso that the politician is 95% sure of this outcome.

Notes: Tutor(s) should shows carefully to students how to read all the table givenin the lectures-room.

0.7 Tutorial 7

1. Given that the margin error is 15 seconds from true mean with 95% confident.Calculate the real expected time if it is normally distributed with standard deviationis 40 seconds.

2. From 100 laundry shops, we observed that 64 from the shops using Brand A’sdetergent.

(a) Find the 90% confidence interval for proportion of the shop using Brand A’sdetergent.

(b) Find the 95% confidence interval for proportion of the shop using Brand A’sdetergent. Find the 99% confidence interval for proportion of the shop usingBrand A’s detergent.

3. The mean and standard deviation for the achievement of randomly selected 25graduates is 15 and 0.3 respectively. Find the 95% confidence interval for the truemean achievement.

4. An automatic machine that can produce juice products with it’s volume followed anormal distribution with the standard deviation 15 ml. What is the 95% confidenceinterval if n = 25 and x = 225.

5. A botanist wants to estimate the minima number of the trees in a jungle near toMalaysia National Park. How many of the trees per acre should be checked if hewants 95% confident that the differences between sample and population means iswithin 3 trees per acre. From previous experience σ2 = 12 trees per acre.

6. In a group of 500 male adults who drink milk, 86 of them like Brand A’s milk. Findthe 90% confidence interval for the proportion of adults like to drink Brand A’smilk.

7. The following data represent the time of previewing the movies by 2 good mannerfilm companies.

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Company A 102 86 98 109 92Company B 81 165 97 134 92

Assumed that both population are normally distributed. Get the 95% confidenceinterval for difference of the mean of the time of casting the movies by the 2 filmcompanies. We found earlier that the variance for both preview times are similar.

8. The data showed the wear resistance measurement for two type of tyres which puton each side of fronts wheels.

Tyre on the right wheel, A 8.8 9.7 9.8 10.6 12.3Tyre on the left wheel, B 8.3 9.1 9.4 10.2 11.8

Find the 95% confidence interval for the mean difference of wear between tyre Aand Tyre B?

9. According to the following information.

Type A 34.8 34.5 35.0 34.6 34.2 34.6 34.9Type B 33.8 35.0 33.5 33.3 34.5 33.1 35.4 33.9 33.9

Find the 95% confidence interval for difference of the mean if

(a) The population variances are similar.

(b) The population variances are assumed to be different.

(c) We know that σ2A = σ2

B = 0.36.

(d) We know that σ2A = 0.49 and σ2

B = 0.36.

(a) A religous scholar from one good university conducted a survey of 400 of itsstudents and found that the average amount of time spent prayer to GOD bytheir student was 12.5 days per month with a standard deviation of 5.4 days.

i. Estimate µ the true mean amount of time spent prayer to GOD by theirstudent in the university and find the margin of error of your estimate.

ii. Construct 99% confidence interval for µ.

(b) It is interest to know if the average time it takes Ambulance A to reach thescene of an accident differs from that of an Ambulance B to reach the sameaccident. The summary data is listed below

Ambulance A Ambulance BnA = 60 nB = 55

xA = 4.2 xB = 4.5

s2A = 0.08 s2B = 0.10

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Estimate the difference in times (in minutes) between Ambulance A and Am-bulance B using 95% confidence interval. Do the times differ? Why?

(c) In UPM, 500 open-minded and deligent students where asked if they would usepublic transportation if a new system was implemented. From 500 students,420 of them say "yes" A proposal for the new public transportation systemwill be submitted to the authority if it can attract at least 80% of the retiredpeople to use public transport.

i. Estimate p the true proportion of retired people that will use public trans-port and find the margin of error of your estimate.

ii. Construct the 90% confidence interval for p. Should the proposal be sub-mitted or not? Why?

(d) A good and generous of an aircraft CEO company wants to estimate the latearrival rate for flights of his company. How many flights must he included in asimple random sample if he wants to be 95% confident that the true populationproportion of flights that arrive late lies within 0.1 of his sample proportion?

0.8 Tutorial 8

1. The manufacturer of the X-15 steel belted radial truck tyre claims that the meanmileage the tyre can be driven before the tread wears out is 60,000 miles. Thestandard deviation of the mileages is known to be 5,000 miles. The SimeX TruckCompany bought 48 tyres and found that the mean mileage for their trucks is 59,500miles.

(a) Is SimeX’s experience different from that claimed by the manufacturer at the0.05 significance level?

(b) What is the p value for the test in (a)?

(Modified from Mason, Lind and Marchal, Statistical Techniques in Business andEconomics)

2. A new halal food outlet claims that the average time between a customer enteringthe outlet and the customer being served is no more than 5 minutes. The standarddeviation of times to service is known to be 1 minute. A simple random sample of25 customers had a mean service time of 5 minutes and 30 seconds.

(a) Does this sample reject the claim made by the outlet? Use a 5% test.

(b) What is the p value for the test in (a)?

(c) What assumptions must be made for this test to be valid? Do you think theseassumptions will be satisfied here?

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3. In a test of the hypotheses:

H0 : µ ≥ 20 versus HA : µ < 20

the reported p value was 0.04.

(a) Is H0 rejected at the 5% level?

(b) Is H0 rejected at the 1% level?

(c) A researcher wants to test the hypotheses

H0 : µ = 20 versus HA : µ 6= 20

at the 5% level. Is H0 rejected?

4. A greedy union official has claimed that the weekly wages paid to domestic workersin Serdang are normally distributed with a mean of RM40 and a standard deviationof RM8. A good economist accepts that weekly wages are normally distributed andthat the standard deviation is RM8 but believes that the quoted figure for the meanis too low. To test the figure for the mean, the good economist plans to take arandom sample of 16 domestic workers and to calculate their mean wage. The goodeconomist then proposes to reject the claim made by the greedy union official if thiscalculated sample mean wage is RM42 or more. If the sample mean wage is lessthan RM42, the good economist will keep quiet and not reject the greedy unionofficial’s claim.

(a) What are the null and alternative hypotheses to be tested here?

(b) If the union official’s claim is in fact true, what is the probability that theeconomist will falsely reject the claim. What type of error is this?

(c) If in fact the population mean wage is RM45 (and so the quoted union figurefor the mean wage is indeed too low), what is the probability that the goodeconomist will not reject the claim? What type of error is this?

(d) What decision rule should the good economist adopt if he wants to have onlya 5% chance of rejecting the union official’s claim when it is true? Specify thisdecision rule in the same way as above, i.e. as a range of sample means forwhich the claim will be rejected. For this decision rule, what is the probabilitythat the economist will not reject the union official’s claim when in fact thepopulation mean wage is RM45?

(e) The good economist would prefer to have a decision rule such that the prob-ability of not rejecting the greedy union official’s claim when it is true will be5% and the probability of not rejecting the claim when the population mean

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is RM45 will be 10%. How large a sample would this require and what wouldthe decision rule be?

5. If 10% of MTH3003 students are vegetarians, test the hypothesis that students whogamble are less likely to be vegetarians. If the 120 students polled, 10 claimed tobe a vegetarian.

6. For randomly selected adults IQ scores are normally distributed with a mean of 100and a standard deviation of 15. A sample of 24 randomly selected college professorsresulted in IQ scores having a standard deviation of 10. Test the claim that the IQscores for college professors is the same as the general population, that is 15. Use a0.05 level of significance. Answer: Reject the null hypothesis

7. A medical researcher wishes to see whether the variances of the heart rates (inbeats per minutes) of smokers are different from the variances of heart rates ofpeople who do not smoke. Two samples are selected, and the data are shown below.Using α = 0.05, is there enough evidence to support the claim?

Smokers Non-smokern1 = 26 n2 = 18

s1 = 6 s2 = 3.16

8. A manufacturer wishes to determine whether there is less variability in the silverplating done by Company 1 than that done by Company 2. Independent randomsamples yield the following results. Do the populations have different variances?solution: reject H0 since 3.14 > 2.82

Sample 1 Sample 2n 12 12s2 0.035 mil 0.062 mil

9. Each respondent in the Current Population Survey of May 2008 was classified asemployed, unemployed, or outside the labor force. The results for men in Serdangage 35-44 can be cross-tabulated by marital status, as follows:

Widowed, NeverMarried divorced, Married

separatedEmployed 679 103 114Unemployed 63 10 20Not in labour force 42 18 25

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Men of different marital status seem to have different distributions of labor forcestatus. Or is this just chance variation? (you may assume the table results from asimple random sample.)

0.9 Tutorial 9

1. What it’s mean by negative linear relationship between x and y? Can you guessabout the slope and r values?

2. You get y = 96.14, what this result means for you?

3. You get y = 2.75 + 96.14x, what this result means for you?

4. You get y = 2.75− 96.14x, what this result means for you?

5. In area of Pasir Mas, Kelantan, records were kept on the relationship between therainfall (in inches) and the yield of paddy (bushel per acre) as follow

Rain fall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0Yield(bushel per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8

(a) Find the least-squares prediction line.

(b) Determine whether there is a significant linear relationship between x and y.Test at the 5% level of significance.

(c) Find a 95% confidence interval estimate of the slope β.

(d) Calculate r and r2

6. A private agency conducted a survey in 9 regions of the country to determine theaverage weekly spending in RM per person on "Nasi lemak", x, and "Teh Tarik",y. The data are listed below.

Region 1 2 3 4 5 6 7 8 9x 12.8 13.20 9.50 10.30 9.80 11.70 10.00 8.90 11.60y 8.50 7.60 6.90 6.80 6.80 5.70 6.50 4.90 7.00

(a) Construct a scatter plot. What do you see?

(b) Find the regression line. Answer: y = 0.449x+ 1.87

(c) Find the coefficient of determination. What can you conclude? Answer: 0.437

(d) Find the standard error of estimate. Answer: 0.8247

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(e) Construct a 95% prediction interval for the weekly spending on Nasi Lemakwhen the amount spent on Teh Tarik is RM9.50. Answer: (3.983,8.279)

7. The data are given below.

Student CGPA Age1 3.5 232 2.8 283 3.9 224 3.4 275 2.3 216 3.3 26

(a) Define x and y.

(b) Find the least-squares prediction line.

(c) Determine whether there is a significant linear relationship between x and y.Test at the 5% level of significance.

(d) Find a 95% confidence interval estimate of the slope β.

(e) Calculate r and r2

(f) Predict the CGPA for 20-year-old student.

0.10 Tutorial 10

1. Identify the factor, its levels, the treatments, the response vari- able, the experi-mental unit, and the observational unit in the following situa- tions:

(a) An agricultural experimental station is going to test two varieties of wheat.Each variety will be planted on 3 fields, and the yield from the field will bemeasured.

(b) An agricultural experimental station is going to test two varieties of wheat.Each variety will be tested with two types of fertilizers. Each combination willbe applied to two plots of land. The yield will be measured for each plot.

(c) Fish farmers want to study the effect of an anti-bacterial drug on the amount ofbacteria in fish gills. The drug is administered at three dose- levels (none, 20,and 40 mg/100L). Each dose is administered to a large controlled tank throughthe filtration system. Each tank has 100 fish. At the end of the experiment, thefish are killed, and the amount of bacteria in the gills of each fish is measured.

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2. Four treatments for fever blisters, including a placebo (A), were randomly assignedto 20 patients. The data below show, for each treatment, the numbers of days frominitial appearance of the blisters until healing is complete

Treatment Number of daysA 5 8 7 7 8B 4 6 6 3 5C 6 4 4 5 4D 7 4 6 6 5

Test the hypothesis, at the 5% significance level, that there is no difference betweenthe four treatments with respect to mean time of healing.

3. Three special micro-oven in a metal working shop are used to heat steel specimens.All the ovens are supposed to operate at the same temperature. It is known that thetemperature of an oven varies, and its is suspected that there are significance meantemperature differences between ovens. The table below shows the temperatures,in degrees centigrade, of reach the three ovens on a random sample of heatings.

Oven Temperature (oC)1 494 497 481 496 4872 489 494 479 4783 489 483 487 472 472 477

Stating any necessary assumptions, test for a difference between mean oven temper-atures.

4. Serdang Health Authority has a policy whereby any patient admitted to a hospitalwith a suspected coronary heart attack is automatically placed in the intensive careunit. The table below gives the number of hours spent in intensive care by suchpatients at five hospitals in Serdang area.

HospitalsA B C D E30 42 65 67 7025 57 46 58 6312 47 55 81 8023 30 2716

Use a one factor analysis of variance to test, at the 1% level of significance, fordifferences between hospitals.

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5. Prior to submitting a quotation for a construction project, companies prepare a de-tailed analysis of the estimated labour and materials costs required to complete theproject. A company which employs three project cost assessors, wished to comparethe mean values of these assessors’ cost estimates. This was done by requiring eachassessor to estimate independently the costs of the same four construction projects.These costs, in RM0000s, are shown in the next column.

AssessorA B C

Project 1 46 49 44Project 2 62 63 59Project 3 50 54 54Project 4 66 68 63

Perform a two factor analysis of variance on these data to test, at the 5% significancelevel, that there is no difference between the assessors’ mean cost estimates.

6. In an experiment to investigate the warping of copper plates, the two factors studiedwere the temperature and the copper content of the plates. The response variablewas a measure of the amount of warping. The resultant data are as follows.

Copper content (%)Temp (oC) 40 60 80 100

50 17 19 23 2975 12 15 18 27100 14 19 22 30125 17 20 22 30

Stating all necessary assumptions, analyse for significant effects.

7. A drug is produced by a fermentation process. An experiment was run to comparethree similar chemical salts, X, Y and Z, in the production of the drug. Since therewere only three of each of four types of fermenter A, B, C and D available for usein the production, three fermentations were started in each type of fermenter, onecontaining salt X, another salt Y and the third salt Z. After several days, sampleswere taken from each fermenter and analysed. The results, in coded form, were asfollows.

Fermenter typeA B C D

X 67 y 73 X 72 Z 70Z 68 Z 65 Y 80 X 68Y 78 X 69 Z 73 Y 69

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State the type of experimental design used. Test, at the 5% level of significance,the hypothesis that the type of salt does not affect the fermentation. Comment onwhat assumption you have made about the interaction between type of fermenterand type of salt.

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0.11 Binomial Table and Standard Normal Table

p

x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50n = 1 0 .9900 .9500 .9000 .8500 .8000 .7500 .7000 .6500 .6000 .5500 .5000

1 .0100 .0500 .1000 .1500 .2000 .2500 .3000 .3500 .4000 .4500 .5000n = 5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1160 .0778 .0503 .0313

1 .0480 .2036 .3281 .3915 .4096 .3955 .3602 .3124 .2592 .2059 .15632 .0010 .0214 .0729 .1382 .2048 .2637 .3087 .3364 .3456 .3369 .31253 .0011 .0081 .0244 .0512 .0879 .1323 .1811 .2304 .2757 .31254 .0005 .0022 .0064 .0146 .0284 .0488 .0768 .1128 .15635 .0001 .0003 .0010 .0024 .0053 .0102 .0185 .0313

n = 8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0319 .0168 .0084 .00391 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .0896 .0548 .03132 .0026 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2090 .1569 .10943 .0001 .0054 .0331 .0839 .1468 .2076 .2541 .2786 .2787 .2568 .21884 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .2322 .2627 .27345 .0004 .0026 .0092 .0231 .0467 .0808 .1239 .1719 .21886 .0002 .0011 .0038 .0100 .0217 .0413 .0703 .10947 .0001 .0004 .0012 .0033 .0079 .0164 .03138 .0001 .0002 .0007 .0017 .0039

n = 10 0 .9044 .5987 .3487 .1969 .1074 .0563 .0282 .0135 .0060 .0025 .00101 .0914 .3151 .3874 .3474 .2684 .1877 .1211 .0725 .0403 .0207 .00982 .0042 .0746 .1937 .2759 .3020 .2816 .2335 .1757 .1209 .0763 .04393 .0001 .0105 .0574 .1298 .2013 .2503 .2668 .2522 .2150 .1665 .11724 .0010 .0112 .0401 .0881 .1460 .2001 .2377 .2508 .2384 .20515 .0001 .0015 .0085 .0264 .0584 .1029 .1536 .2007 .2340 .24616 .0001 .0012 .0055 .0162 .0368 .0689 .1115 .1596 .20517 .0001 .0008 .0031 .0090 .0212 .0425 .0746 .11728 .0001 .0004 .0014 .0043 .0106 .0229 .04399 .0001 .0005 .0016 .0042 .009810 .0001 .0003 .0010

Table 4: The table gives the values for Pr(X = x) = b(x;n, p)

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z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

Table 5: Standard Normal: The values in the table are the areas between zero and thez-score. That is, Pr(Z < z − score)

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Conf. Level 50% 80% 90% 95% 98% 99%One Tail 0.250 0.100 0.050 0.025 0.010 0.005Two Tail 0.500 0.200 0.100 0.050 0.020 0.010

df . . . . . .1 1.000 3.078 6.314 12.706 31.821 63.6572 0.816 1.886 2.920 4.303 6.965 9.9253 0.765 1.638 2.353 3.182 4.541 5.8414 0.741 1.533 2.132 2.776 3.747 4.6045 0.727 1.476 2.015 2.571 3.365 4.0326 0.718 1.440 1.943 2.447 3.143 3.7077 0.711 1.415 1.895 2.365 2.998 3.4998 0.706 1.397 1.860 2.306 2.896 3.3559 0.703 1.383 1.833 2.262 2.821 3.25010 0.700 1.372 1.812 2.228 2.764 3.16911 0.697 1.363 1.796 2.201 2.718 3.10612 0.695 1.356 1.782 2.179 2.681 3.05513 0.694 1.350 1.771 2.160 2.650 3.01214 0.692 1.345 1.761 2.145 2.624 2.97715 0.691 1.341 1.753 2.131 2.602 2.94716 0.690 1.337 1.746 2.120 2.583 2.92117 0.689 1.333 1.740 2.110 2.567 2.89818 0.688 1.330 1.734 2.101 2.552 2.87819 0.688 1.328 1.729 2.093 2.539 2.86120 0.687 1.325 1.725 2.086 2.528 2.84521 0.686 1.323 1.721 2.080 2.518 2.83122 0.686 1.321 1.717 2.074 2.508 2.81923 0.685 1.319 1.714 2.069 2.500 2.80724 0.685 1.318 1.711 2.064 2.492 2.79725 0.684 1.316 1.708 2.060 2.485 2.78726 0.684 1.315 1.706 2.056 2.479 2.77927 0.684 1.314 1.703 2.052 2.473 2.77128 0.683 1.313 1.701 2.048 2.467 2.76329 0.683 1.311 1.699 2.045 2.462 2.75630 0.683 1.310 1.697 2.042 2.457 2.75040 0.681 1.303 1.684 2.021 2.423 2.70450 0.679 1.299 1.676 2.009 2.403 2.67860 0.679 1.296 1.671 2.000 2.390 2.66070 0.678 1.294 1.667 1.994 2.381 2.64880 0.678 1.292 1.664 1.990 2.374 2.63990 0.677 1.291 1.662 1.987 2.368 2.632100 0.677 1.290 1.660 1.984 2.364 2.626z 0.674 1.282 1.645 1.960 2.326 2.576

Table 6: Student’s t Probabilities: The values in the table are the areas critical values forthe given areas in the right tail or in both tails.

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df 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.0051 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.8792 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.5973 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.8384 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.8605 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.7506 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.5487 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.2788 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.9559 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.58910 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.18811 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.75712 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.30013 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.81914 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.31915 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.80116 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.26717 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.71818 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.15619 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.58220 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.99721 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.40122 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.79623 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.18124 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.55925 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.92826 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.29027 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.64528 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.99329 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.33630 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.67240 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.76650 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.49060 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.95270 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.21580 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.32190 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169

Table 7: Chi-Square Probabilities: The areas given across the top are the areas to theright of the critical value. To look up an area on the left, subtract it from one, and thenlook it up (ie: 0.05 on the left is 0.95 on the right)

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