CT Statistics and Correlation 1 · The Mean, the Median and the Mode 1. The mean, the median and...

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1CHAPTER

Skill Builder: The Mean, the Median and the Mode 13

1 Find the mean, median, and mode of the following set of numbers:

{ 1, 1, 1, 2, 2, 7, 9, 12, 16 }

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{ 10, 15, 20, 20, 22, 24, 25, 30 }

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{ 2, 6, 1, 2, 1, 2, 5, 4, 2, 3 }

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{ 1, 4, 1, 3, 1, 1, 2, 1, 3, 2 }

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5 The following set S = { 2, 4, 7, 7, 9, 10, 10, 12 } is missing one of its numbers. If we know that the mode is 10, what is the missing number?

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6 The following set S = { 1, 1, 1, 3, 3, 7, 9, 10 } is missing one of its numbers. If we know that this set has 2 modes, what is the missing number?

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1 The Mean, the Median and the Mode REVIEW

Statistics and Correlation

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14 Chapter 1: Statistics and Correlation

7 The following set S = { 5, 10, 14, 18, 20, 24, 25 } is missing a number. If we know that the median is 17, what is the missing number?

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8 The following set S = { 1, 2, 5, 7, 10, 14 } is missing one of its numbers. If we know that 50% of the numbers in the set are less than 6, what is the missing number?

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9 The average free-throw shooting percentage for the starting 5 players on the Riverdale High School basketball team is 72%. When the newspaper printed the team’s statistics, only four out of the five starters percentages were readable. If the four readable percentages were 62, 68, 74 and 80, what was the fifth percentage that was too smudged to read?

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10 Tanya has six pairs of shoes in her closet. She knows than the average cost of one of the pairs of shoes is $45. She has receipts for 5 of the 6 pairs. If her receipts total $215, what did the sixth pair of shoes cost?

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MHS Critical Thinking and Big Ideas 15

CRITICAL THINKING AND BIG IDEAS

The Mean, the Median and the Mode

1. The mean, the median and the mode are measures of central tendency. An outlier is a value in a data set that is far away from the main group of data. Please describe each of these measures of central tendency below and comment on which one will be affected the most by an outlier in the data set.

a. The Mean:

b. The Median:

c. The Mode:

d. Which will be affected the most by an outlier and why?


16 MHS Critical Thinking and Big Ideas

2. Provide an algebraic rule for determining the median of a data set. Use the data sets provided to illustrate how your rule works. Does it make a difference whether the number of values is even or odd?

• Odd Set: {2, 8, 10, 12, 16, 16, 18, 20, 25, 27, 28, 28, 30}

• Even Set: {40, 42, 46, 53, 58, 60, 66, 69, 74, 78, 80, 86}

Show all work:

Algebraic Rule:

Finding the median of each set:

Does it make a difference whether the number of values is even or odd?

Historical Note:The “median” showed up in the Talmud scripts in the 13th century and later, in Edward Wright’s 1599 book on Navigation.



3. Naomi is learning about statistics in her math class. She has just learned about the three measures of central tendency; the mean, the median and the mode. She isn’t sure why there are three different types of measures of central tendencies; so she texts her older sister, Ramona, who is studying statistics in University. Please help out by writing the detailed explanation that Naomi is asking for.

I don’t understand, can you write a detailed explanation?

It depends on what the data in the question looks like.

Which is the best measure of central tendency to use?

Sure, let me know what you need help with.

Hi Ramona, I need help with statistics. Can you help?

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1 Organize the following group of data into a stem-and-leaf plot.

17, 23, 24, 37, 45, 52, 58, 58, 73

2 Organize the following group of data into a stem-and-leaf plot.

10, 11, 12, 19, 23, 27, 31, 38, 39, 40, 42, 57

3 Given the stem-and-leaf plot below, list the data values in increasing order.

4 Given the stem-and-leaf plot below, list the data values in increasing order.

5 Organize the following two groups of data with a double stem-and-leaf plot.

Group A

11, 12, 12, 17, 42, 43, 52, 55, 61, 63, 72

Group B

25, 27, 32, 32, 32, 34, 41, 42, 47, 49, 61, 67

6 Organize the following two groups of data with a double stem-and-leaf plot.

Group A

22, 24, 36, 42, 44, 45, 53, 53, 55, 67, 68, 68

Group B

17, 18, 21, 22, 29, 41, 43, 45, 45, 49, 61, 62, 64

2 Stem-and-Leaf Plots EVALUATED

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Skill Builder: Stem-and-Leaf Plots 19

7 Given the following stem-and-leaf plot, determine the range for this group of data values.

8 Given the following stem-and-leaf plot, determine the median for this group of data values.

9 Given the following double stem-and-leaf plot, determine which group has the higher mean.

10 Given the following double stem-and-leaf plot, determine which group has the higher median.




Stem-and-Leaf Plots

1. Describe how to read a stem and leaf plot. Use the provided example to illustrate your explanation.

Description

2. Your friend Stanley missed math class today and does not know how to read a back-to-back or double-sided stem and leaf plot like the one below. Please write a detailed explanation to teach Stanley how to read this type of plot.

Heart RatesGroup A Stem Group B8, 5, 4 119, 8, 7, 7, 6, 5, 5, 5, 1 12 4, 4, 66, 4, 3, 3, 1, 0 13 2, 2, 3, 4, 5, 5, 6, 93, 2, 0 14 1, 1, 5, 7

15 016 1, 8

Explanation



3. Using the following stem and leaf plot, outline the steps necessary to calculate the mean, the median, the mode and the range of the data set and then perform these calculations.

PointsStem Leaf0 3, 4, 4, 61 0, 0, 4, 6, 7, 7, 9, 92 1, 4, 93 0, 0, 0, 2, 4, 84 4, 6, 7, 9, 95 0, 0

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1 In any distribution, what percentage of data is greater than the 90th percentile?

2 What percentile rank is in between the 97th and 98th percentiles?

3 Given the list of data values below, what is the percentile rank of 61?

49, 52, 55, 61, 72, 74, 75, 76, 76, 77, 81, 85, 87, 91

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4 Given the list of data values below, what is the percentile rank of 9?

2, 5, 5, 7, 8, 9, 9, 9, 10, 11, 15

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5 Given the list of data values below, which one has a percentile rank of at least 65?

45, 47, 48, 51, 52, 52, 55, 59, 60, 61, 63, 65, 65

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6 Given the list of data values below, which one has a percentile rank of at least 84?

1, 1, 2, 3, 3, 4, 5, 6, 7, 7, 7, 8, 9, 10, 11, 12, 12, 13, 15

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3 Percentiles and Calculating Percentile Rank EVALUATED

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Skill Builder: Percentiles and Calculating Percentile Rank 27

7 The data below shows the height, in centimetres, of 300 elementary school children.

⎫ ⎬ ⎭ ⎫ ⎬ ⎭59, …, 80, 81, 81, 81

90 children 210 children

82, 83, …, 121

What percentile rank should be assigned to a child whose height is 82cm?

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8 270 students wrote a grade 10 final Math exam. What was Dean’s percentile rank if:

• 204 students obtained a lower test score than Dean.

• 65 students obtained a higher test score than Dean.

• No one obtained the same test score as Dean.

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9 The data below shows the results of the 400 students at McGill University who wrote a Calculus exam.

⎫ ⎬ ⎭78, 78, 78, 78, 79, … 100

240 students⎫ ⎬ ⎭

160 students

45, 46, … 77

Tim’s result was 78. What was his percentile rank?

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10 The data below displays the daily pay of 679 workers at an oil drilling site. All of the values are in dollars.

⎫ ⎬ ⎭

460 people

375, … 470 ⎫ ⎬ ⎭

6 people

475, … 475 ⎫ ⎬ ⎭

213 people

480, … 725

What is the percentile rank of a worker earning $475?

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Percentiles and Calculating Percentile Rank

1. Exploring Percentiles

a. What is the purpose of dividing values into percentile ranks?

b. Provide the formula for calculating the percentile rank.

c. Why do you believe that you always round a percentile up to the next highest integer when calculating percentiles?

d. Describe how to work backwards to calculate which value occupies a certain percentile.



2. Kaya, Thalia and Mathilde are all in the same math class. The results of their mid-year exam were posted on their teacher’s classroom door using a stem and leaf plot.

a. Kaya’s grade was equal to the median of the class. What was her grade?

b. In what percentile is Kaya’s grade?

c. By definition, the median is the middle value. How is it possible that Kaya’s grade does not have a percentile rank of 50?

d. Thalia’s grade is 7.4% higher than the mean average. What is Thalia’s percentile rank?

e. Mathilde’s mark is in the 73rd or 74th or the 75th percentile. What is Mathilde’s mark?

f. How is it possible that Mathilde’s mark can be associated with three different percentile ranks?



3. Athletics: Students always like to try to stump their math teacher with riddles. Charlie put up his hand in class and asked the teacher, “If you’re running a race with 100 runners and you pass the runner in second place, what place are you in?” The teacher responded, “I know you’re hoping that I say first place, but that would be wrong. If you pass the runner in second place, you are in second place. Now it’s my turn to ask you a question.”

The teacher then asked the class the question, “You are in a race with 89 other runners. You are currently tied with 3 other runners and there are 76 runners behind you. What is your current percentile rank?”

Show all work

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1 Calculate the mean of the following list of data values.

3, 5, 8, 10, 14, 16

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2 Calculate the mean of the following list of data values.

20, 40, 50, 70, 80, 85, 90, 95

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3 Calculate the value of the following expression.

|70 – 30| = ?

4 What is the absolute value of 2 – 5?

5 By completing the table below, determine the value of ∑xi, ∑(xi – x), and ∑|xi – x|.

i xi xi – x |xi – x|1 2 –8.6 8.62 7 –3.6 3.63 11 0.4 0.44 15 4.4 4.45 18 7.4 7.4

Total

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6 By completing the table below, determine the value of ∑xi, ∑(xi – x), and ∑|xi – x|.

i xi xi – x |xi – x|1 10 –15 152 20 –5 53 30 5 54 40 15 15

Total

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4 The Mean Deviation EVALUATED

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Skill Builder: The Mean Deviation 37

7 Given the table below, calculate the Mean Deviation.

i xi xi – x |xi – x|1 2 –8.6 8.62 7 –3.6 3.63 11 0.4 0.44 15 4.4 4.45 18 7.4 7.4

Total 53 0 24.4

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8 Given the table below, calculate the Mean Deviation.

i xi xi – x |xi – x|1 10 –15 152 20 –5 53 30 5 54 40 15 15

Total 100 0 40

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9 Calculate the Mean Deviation of the following list of data values.

1, 3, 5, 6, 10, 11

10 Given the two tables below, determine which Math class had the more homogeneous distribution of marks by using the Mean Deviation.

Mr. Cooper’s Class

i xi xi – x |xi – x|1 60 –16 162 70 –6 63 75 –1 14 85 9 95 90 14 14

Total 380 0 46

Mr. Dean’s Class

i xi xi – x |xi – x|1 60 –15 152 65 –10 103 70 –5 54 75 0 05 80 5 56 100 25 25

Total 450 0 60

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The Mean Deviation

1. The mean average is a measure of central tendency, as are the median and the mode. All three give us information about the centre of the distribution we are looking at. The mean deviation is a measure of dispersion. Rather than telling us where the centre is, it indicates how far values are from the centre. Use the given set of numbers to answer the following questions.

{ }1, 1, 1, 1, 5, 5, 5, 5A =

a. How many values are there?

b. Calculate the mean:

c. How far is each value in the distribution away from the mean? (Remember that distances are always positive).

d. Find the sum of each value’s distance from the mean (take the answer from the previous question and multiply it by the number of values).

e. If you divide by the number of values, what do you get?

f. This tells us that the average distance from the mean for any value in this distribution is .

This was very easy to calculate in this case, because the only values in the distribution were 1 and 5. It does however; give us a good idea of what the Mean Deviation represents, the average distance of any value in the distribution from the mean.

Try the same thing again with { }8, 10, 12, 14, 16, 18, 20S =

g. How many values are there?

h. Calculate the mean:

i. How far is each value in the distribution away from the mean? (Remember that distances are always positive).

8 = 10 = 12 = 14 = 16 = 18 = 20 =



j. Find the sum of the answers to the previous question

k. If you divide by the number of values, what do you get? l. This tells us that the average distance from the mean, for any value in this

distribution is .

2. The formula to calculate the Mean Deviation is given below. Please explain what the formula is asking the user to do, in words.

Mean Deviation=ix x

n

Σ −

Meaning of Σ:

Meaning of xi:

Meaning of x:

Meaning of n:

Meaning of ||:

Explanation of what the formula is asking the user to do:



3. Mr. Chips teaches two math classes who wrote the same exam last week. The exam results of each class, as percentages, are listed below.

Class A Class B56, 59, 60, 64, 66, 66, 72,80, 82, 82, 85, 86, 89, 90

21, 60, 65, 66, 67, 70, 75, 76,80, 83, 84, 84, 86, 87, 88, 90

a. Identify which class is more homogenous (evenly distributed) by calculating the mean deviations.

Class A Class B

Mean Deviation= ix x

n

Σ −

i ix ix x− ix x−

1234567891011121314

Total

Mean Deviation= ix x

n

Σ −

i ix ix x− ix x−

12345678910111213141516

Total

Which group is more homogenous?

b. If you remove the outlier from Class B (the grade of 21%), does this affect your findings?

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Correlation, Scatter Plots, and the Linear Correlation Coefficient 43

1 Given the following situation, indicate whether the variables have positive, negative, or zero correlation.

“The number of hours chatting on Facebook the night before an exam and the result on that exam.”

2 Given the following situation, indicate whether the variables have positive, negative, or zero correlation.

“The number of push-ups you can do and your I.Q. test score.”

3 Which scatter plots display a positive correlation?

A.

B.

C.

D.

E.

F.

G.

H.

4 Determine which of the scatter plots display negative correlation. Identify the ones with the strongest and weakest negative correlations.

A. B. C.

D. E. F.

5 The ages of the husband and wife for 24 couples are recorded in the scatter plot below.

a. Draw a rectangle that fits around all of the data points.

b. Draw a line through the middle of the rectangle to show an approximate linear correlation for the scatter plot.

c. What kind of correlation is present between the wife’s and husband’s ages?

5 Correlation, Scatter Plots, and the Linear Correlation Coefficient EVALUATED

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6 The arm span and height of 24 people are recorded in the scatter plot below.

a. Draw a rectangle that fits around all of the data points.

b. Draw a line through the middle of the rectangle to show an approximate linear correlation for the scatter plot.

c. What kind of correlation is present between the arm span and height?

7 Given the following list of Linear Correlation Coefficients, determine which one would indicate the strongest correlation.

0.75, 0.9, –0.4, –0.85, –0.7

8 Given the following list of Linear Correlation Coefficients, determine which one would indicate the strongest correlation.

–0.2, 0.7, 0.65, –0.8 , 0.1

9 Given the following list of Linear Correlation Coefficients, determine which one would indicate the weakest correlation.

0.5, –0.8, –0.1, 0.2, 0.7

10 Given the following list of Linear Correlation Coefficients, determine which one would indicate the weakest correlation.

–1 , 0.2 , –0.1 , 0 , 0.8




Correlation, Scatter Plots and the Linear Correlation Coefficient

1. If two variables have a correlation, it means that they fluctuate together. That is, changes in one have a direct effect on the other.

• Positive Correlation: If when one variable increases, the other also increases, the variables have a positive correlation.

• Negative Correlation: If when one variable increases, the other decreases, the variables have a negative correlation.

For each of the following real-world examples, state whether you think they have a positive, negative or no correlation.

a. The number of hours someone spends in the sun and the severity of sunburn.

b. The temperature outside and the number of layers of clothing people wear.

c. Amount of time spent studying and test grades.

d. Number of hours per day spent exercising and hair growth.

e. The outside temperature and bathing suit sales.

f. Number of years of education and incarceration rate (the rate at which people are sentenced to prison terms).

g. Hours of daylight and electricity usage.

h. Hours of exercise and percentage of body fat.

i. Shoe size and grades on a math test.

j. Speed a car travels and fuel consumption.

k. A person’s age and the length of time they can stand on one foot.

l. Number of years spent smoking cigarettes and life expectancy.



Please create a real-world example of your own of each type of correlation.

Positive Correlation: Negative Correlation: Zero Correlation:

2. Exploring the Correlation Coefficient r

a. Plot points on each scatter plot that you believe will result in the indicated

correlation coefficient.y

x

y

x

y

x

y

x

y

x

1r = − 0.75r ≈ − r = 0 r ≈ 0.75 r = 1

b. Express your knowledge of correlation by filling in the blanks below.

• The correlation coefficient r is a value that varies between ≤ r ≤ .

• If r is equal to –1, the data set forms a , with a slope.

• If r is equal to 0, the data set has no .

• If r is equal to 1, the data set forms a , with a slope.

• The correlation coefficient is a measure of how well the data would fit into a mathematical model. A linear function in the form y mx b= + can be used to represent the line of best fit. The stronger the correlation, the all of the data points will be to the line.

3. Meteorology: Halifax, Nova Scotia and Christchurch, New Zealand are roughly the same distance from the equator, but in opposite hemispheres. Let’s investigate the correlation between the temperature data of the two cities. The following table represents the average monthly high temperatures in each city. Fill in the blanks and perform the tasks required below.

Average Monthly High Temperatures (°C)Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecHalifax 1 0 2 5 9 14 17 19 18 13 9 4

Christchurch 22 22 20 17 14 12 11 12 15 17 19 21



a. Plot the data points on the coordinate plane provided. Let x represent the temperatures in Halifax and y represent the temperatures in Christchurch.

b. Is there a correlation between the temperatures of the two cities? . If so, is the correlation positive or negative?

c. As the temperatures in Halifax , they in Christchurch.

d. Draw a line through your data points, trying to keep half of the points above your line and half of them below.

e. Identify the coordinates of two points on the line you have drawn: ( , ) and ( , ).

f. Find the equation of the line that passes through those two points. Recall that to find the equation of a line in the form y mx b= + , calculate the slope and then substitute a point into the equation to solve for the y-intercept. The equation of the line is:

g. The equation you have found is called the line of best fit. It is the mathematical model that this data correlates to.

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1 Given the scatter plot below, what is the equation you should use to estimate the Linear Correlation Coefficient?

2 Given the scatter plot below, what is the equation you should use to estimate the Linear Correlation Coefficient?

3 Given the scatter plot below, determine whether the Linear Correlation Coefficient will be positive or negative, and whether it will be weak or strong.

4 Given the scatter plot below, determine whether the Linear Correlation Coefficient will be positive or negative, and whether it will be weak or strong.

5 Identify the outlier in the scatter plot below.

6 Identify the outlier in the scatter plot below.

6 Estimating the Linear Correlation Coefficient EVALUATED

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Skill Builder: Estimating the Linear Correlation Coefficient 49

7 Estimate the Linear Correlation Coefficient for the scatter plot below and determine whether the correlation is weak or strong, and whether it is positive or negative.







Estimating the Linear Correlation Coefficient

1. Entrepreneurship: Marcus is planning the variety show at his high school this year and wants to set the price so that the show brings in high revenue and is well attended. Marcus is not sure whether the ticket price has any effect on the number of people who come to events. In order to figure it out, he has made a table of values for the last several events at the school and is trying to establish the correlation between ticket price and attendance.

Event Attendance by Ticket PriceTicket ($) 7 6 9 8 10 6.50 7.50 5 8

Attendance 225 270 210 200 175 260 220 280 210

Please help Marcus determine how strong the correlation is between ticket price and attendance.

a. Plot each data point on the coordinate plane provided.

b. Draw a rectangle as tightly around the points as you can.

c. Measure the short side of the rectangle: b =

d. Measure the long side of the rectangle: a =

e. Is the correlation positive or negative?



f. Use the formula 1b

ra

≈ ± −

to determine the correlation coefficient. Note: if

the correlation is positive use a + before the parentheses and if the correlation is negative, use a – before the parentheses.

g. The value of the correlation coefficient is r = .

h. What would you tell Marcus about how ticket price affects attendance?

2. For each of the following, estimate the linear correlation coefficient.

a = ________

b = ________

r = ________

a = ________

b = ________

r = ________

a = ________

b = ________

r = ________

a = ________

b = ________

r = ________

a = ________

b = ________

r = ________

a = ________

b = ________

r = ________



3. Aeronautics: Rosanna is interested in the Aircraft Maintenance program at her local college and needs to write an entrance exam. She was sent a study guide to help her prepare for the test; the following is the first question in her guide.

An aircraft engineer needs to know whether there is a relationship between the wingspan and the length of an aircraft. The diagram below illustrates the length and wingspan of six aircraft in the Air Canada fleet.

a. Make a scatter plot for the lengths and wingspans of planes in the Air Canada fleet.

b. Use the graphical method to determine the correlation coefficient between length

and wingspan: r = .

c. Describe the correlation in words:

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Skill Builder: Calculating the Linear Regression Line 53

1 For the scatter plot below, determine whether the line of best fit is linear or non-linear.

2 For the scatter plot below, determine whether the line of best fit is linear or non-linear.

3 Given the scatter plot below, calculate the slope of the linear regression line.

4 Given the scatter plot below, calculate the slope of the linear regression line.

5 Given the scatter plot below, determine the equation of the linear regression line.

6 Given the scatter plot below, determine the equation of the linear regression line.

7 Calculating the Linear Regression Line EVALUATED

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7 Given the table of values below, use a scatter plot to determine the equation of the linear regression line.

8 Given the table of values below, use a scatter plot to determine the equation of the linear regression line.

x 2 2 3 3 3 3 4 4 4 4 5 5 5y 11 10 11 10 9 8 10 9 8 7 8 7 6

x 5 6 6 6 7 7 7 7 8 8 8 8 9 9y 5 6 5 4 5 4 3 2 4 3 2 1 2 1

9 Given the scatter plot below, predict the value of y when x = 3 by using a linear regression line.

10 Given the scatter plot below, predict the value of y when x = 5 by using a linear regression line.




Calculating the Linear Regression Line

1. Agriculture: Bernie owns a farm in Huntington, Quebec and is considering harvesting some of his pine trees in one section of his property for lumber. He remembers planting many of the trees himself and has been told that his grandfather planted the oldest pine tree 75 years ago.

Bernie has written out the age and heights of some of the trees that he remembers planting.

Age of Pine Tree in Years 8 9 10 12 14 16 18 18 19 20 20Height of Tree in Metres 4 4.6 5.4 5.4 5.8 6 6.1 6.8 7.1 8 8.6

Bernie knows that there is a strong, positive correlation between the age of a pine tree and its height. Given the information provided about the age and heights of the other trees, how tall could Bernie expect the tree his grandfather planted to be?

Use the linear regression line to determine the approximate height of that tree.



2. Foodservice: Lorenzo is a waiter at a fine dining restaurant. Given that his income is highly dependent on the gratuities (tips) received, he always keeps track of them. He recognizes that there is a strong relationship between his total food and beverage sales made during a shift and the amount of tips he receives. The following table illustrates his total food and beverage sales and tips received for his last 8 shifts.

Sales ($) Tips ($)420 85600 110680 140750 155900 180900 1751000 2151050 200

This Saturday is Valentine’s Day and one of the restaurant’s busiest nights of the year. Lorenzo’s boss has asked him to work and told him that he will probably earn at least $300 in tips.

Using a linear regression line, determine how much Lorenzo would need to sell to earn over $300 in tips this Saturday.



3. Anthropology: Temperance is a criminal anthropologist; she works for the local police department, as well as taking cases for the Canadian Security Intelligence Service (CSIS). She was just contacted about bones that were found during excavation for a housing development. When she arrived at the scene, she quickly concluded that the bones were female.

She would like to put together a profile of what this woman looked like, which involves determining height. Temperance found that the femur bone was completely intact. She has a table of other cases she has worked that relates the length of female subjects’ femur bones to their height.

a. Determine the equation of the line of best fit.

b. Given that the length of the femur bone found is 47 cm long, how tall would Temperance expect the subject to have been?

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1 The table of values shows the average mass of various types of cats and the average length from nose to tail. Draw a scatter plot of the data and use it to find the median points: M1, M2 and M3.

Type of Cat Mass (kg) Length (cm)

Lion 180 300

Lioness 140 270

Cheetah 45 180

Cougar 90 240

Jaguar 140 260

Leopard 70 265

Tiger 190 270

Tigress 135 240

Lynx 30 90

2 This table shows the number of days absent from science class and the report card marks for 14 students. Draw a scatter plot of the data. Use the Mayer Line Method to find the two mean points, m1 and m2.

Days Absent Mark (%)

2 810 7312 507 631 7722 3810 443 953 564 718 670 789 6115 40

Leng

th (c

m)

Mass (kg)

Mar

k (%

)

Days Absent

8 Median-Median and the Mayer Line Methods OPTIONAL METHOD

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Skill Builder: Median-Median and the Mayer Line Methods 59

3 Tanya measured the depth of water in a bathtub at two-minute intervals after the tap was turned on. The table shows her data. Find M1, M2, and M3 using the Median-Median Line method.

Time (minutes)

Depth(in cm)

2 7 4 8 6 13 8 1910 2012 2414 3216 3718 3820 4122 4724 49

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4 The scatter plot shows the height and the foot length of each student in a class. Find M1, M2, and M3 and plot the Median-Median line.

Height (cm)

Foot

Len

gth

(cm

)

My Calculations

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5 The body weight and brain weight for 12 small mammals is shown in the table.

a. Make a scatterplot of the data.

b. Draw the Median-Median line and find its equation.

body weight (g)

brain weight (g)

1400 181600 101700 62000 122500 123000 243300 263400 443600 224200 584200 504300 40

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6 This scatter plot identifies the age, in years, of 12 couples who had applied for a marriage license during a one month period.

Use the Median-Median method to find the equation of the regression line.

Husband’s Age

Wife

’s A

ge

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Skill Builder: Median-Median and the Mayer Line Methods 61

7 This table shows the arm span and height of 12 players on a soccer team. Order the data to find m1 and m2 using the Mayer Line method.

Arm Span (cm)

Height (cm)

156 162177 173159 162178 178161 162188 184165 166190 188170 167196 184173 176200 186

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8 An ice cream store created a scatter plot relating the average daily temperature to their ice cream sales. Find the two mean points, m1 and m2, and the equation of the Mayer line.

Average Temperature (ºC)

Ice

Cre

am S

ales

($)

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9 The data set relates the number of chirps per second for striped ground crickets to the temperature in degrees Celsius.

a. Make a scatterplot of the data.

b. Draw the Mayer Line and find its equation.

Temperature (°C) Chirps per second

20 3821 3722 4423 4624 4225 4826 5127 5028 5029 5530 5431 60

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10 The scatter plot contains the weight and blood glucose level of 12 patients. Use the Mayer Line method to find the regression line and then find the equation that describes it.

Weight (kg)G

luco

se (m

mol

/L)

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Median-Median and the Mayer Line Methods

1. Nutrition: Darren is designing a new deli sandwich for the Master Subs franchise where he is the manager. He needs to supply the nutritional information, before his boss will allow him to proceed with testing out the sandwich.

On the company’s website he found the following table outlining the grams of fat and number of calories in each of their current sandwiches.

Sandwich Grams of Fat Total CaloriesVeggie Delight 2.5 230Turkey Breast 3.5 280Black Forest Ham 4.5 290Master Subs Club 4.5 310Oven Roasted Chicken 5 320Roast Beef 5 340Steak & Cheese 10 380Meatball Marinara 18 480Spicy Italian 24 480Chicken & Bacon Ranch 30 610

From listing out the ingredients and using a calories calculator, Darren has figured out that his sandwich will contain 365 calories. He believes that there is a positive correlation between the number of grams of fat and total calories. Follow the steps below to help Darren estimate the grams of fat that his new sandwich will contain, using the Median-Median line method.

a. Divide the ordered pairs in the table into 3 approximately equal groups (try to make sure that the first and last groups contain the same number of pairs).



b. Find the median x-value and median y-value of each group and label them M1, M2, M3.

M1 = (_______, _______)

M2 = (_______, _______)

M3 = (_______, _______)

c. Calculate the slope of (M1, M3).

d. Find point P by taking the mean of the x-values and the mean of the y-values of M1, M2, M3.

P = (_______, _______)

e. The Median-Median line is parallel to 1 3M M and it passes through point P. Determine the equation of the line.

Equation of the line: ____________________________

f. Use the linear regression line to interpolate the number of grams of fat in Darren’s new sandwich.

Estimated grams of fat: ____________________________



2. Transportation: Marcelle is a logistics coordinator for a large pharmaceutical company. Part of her job portfolio consists of managing the travel budgets of the company’s sales force.

Many of Marcelle’s salespeople need to do a lot of driving, usually in rental cars, to visit customers. She recognizes that there is a strong correlation between distances travelled and total transportation cost. Marcelle would like to come up with a formula to help her estimate the total transportation cost of one-day sales trips in advance. She has kept track of the following data from recent one-day trips to help her with this task.

Distance (Km) 140 180 190 200 200 220 225 240 255Cost ($) 85 90 95 100 110 110 115 120 125

Follow the steps below to help Marcelle come up with a formula (linear regression line) to estimate travel costs, using the Mayer Line Method.

a. Divide the data into two equal groups. If there are an odd number of values, place the extra one in the group that has points closest to it.

b. Find the “mean points” by finding the mean of the x-values and the mean of y-values in each group.

c. The Mayer line passes through these two points. Find the equation of the Mayer Line.

d. Use the Mayer Line to determine the estimated travel cost for a one-day sales trip where the salesperson travelled 450 kilometres.

e. Why do you believe that this formula will only work for one-day sales trips and not multiple-day trips?



3. Kinesiology: Renée is a professor in the Kinesiology department at McGill University. One of her students is writing a research paper on the body mechanics of high performance distance runners. The Montreal Marathon is run every September around Parc Jean-Drapeau and certain sectors of the city.

Her student was at the race and timed the speed that the top 15 runners were travelling at, as well as the steps they took per second. He compiled the information in the following scatter plot.

Renée would like her student to figure out the correlation between a runner’s speed and the number of steps. To do so, Renée would like him to use all three methods:

i. The line of best fit.

ii. The Median-Median Line.

iii. The Mayer Line.

Please help Renée’s student with his research, by calculating the rule of the linear regression line using all three methods.



Show all Work:

Line of Best Fit: ________________________________

Median-Median Line: ________________________________

Mayer Line: ________________________________

Did you get the same answer with all three methods?

If not, why do you believe you have a different result for each method?

Which method do you prefer? Remember that on the final exam you can choose which method you want to use to answer correlation questions.

Date post:	23-Aug-2020
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CT Statistics and Correlation 1 · The Mean, the Median and the Mode 1. The mean, the median and...

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