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Chapter 12: Linear Regression and Correlation

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 12: Linear Regression and Correlation

43 problems · 7 figures

12-1 §12.1 source ↗

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

What are the dependent and independent variables?

Answer

dependent variable: fee amount; independent variable: time

12-3 §12.1 source ↗

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

Graph the equation from Exercise 12.2.

Answer

Figure 12.33 in the book ↗

12-5 §12.1 source ↗


Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.

Graph the equation from Exercise 12.4.

Answer

Figure 12.34 in the book ↗

12-7 §12.1 source ↗


Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.

Which of the following equations are linear?

a. y = 6x + 8

b. y + 7 = 3x

c. y – x = 8x2

d. 4y = 8

Answer

y = 6x + 8, 4y = 8, and y + 7 = 3x are all linear equations.

12-9 §12.1 source ↗

Table 12.13 contains real data for the first two decades of flu reporting.

Use the columns "year" and "# flu cases diagnosed. Why is “year” the independent variable and “# flu cases diagnosed.” the dependent variable (instead of the reverse)?

Table 12.13 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

The number of flu cases depends on the year. Therefore, year becomes the independent variable and the number of flu cases is the dependent variable.

12-11 §12.1 source ↗


Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.

What is the y-intercept and what is the slope? Interpret them using complete sentences.

Answer

The y-intercept is 50 (a = 50). At the start of the cleaning, the company charges a one-time fee of $50 (this is when x = 0). The slope is 100 (b = 100). For each session, the company charges $100 for each hour they clean.

12-13 §12.1 source ↗


Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.

How many pounds of soil does the shoreline lose in a year?

Answer

12,000 pounds of soil

12-15 §12.1 source ↗


Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x where x is the number of hours passed in an eight-hour day of trading.

What are the slope and y-intercept? Interpret their meaning.

Answer

The slope is –1.5 (b = –1.5). This means the stock is losing value at a rate of $1.50 per hour. The y-intercept is $15 (a = 15). This means the price of stock before the trading day was $15.

12-17 §12.2 source ↗

Does the scatter plot appear linear? Strong or weak? Positive or negative?
Figure 12.28 in the book ↗

Answer

The data appear to be linear with a strong, positive correlation.

12-19 §12.2 source ↗

Does the scatter plot appear linear? Strong or weak? Positive or negative?
Figure 12.30 in the book ↗

Answer

The data appear to have no correlation.

12-21 §12.3 source ↗

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

Use regression to find the equation for the line of best fit.

Table 12.14 in the book ↗
x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Answer

ŷ = 2.23 + 1.99x

12-23 §12.3 source ↗

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

What is the slope of the line of best fit? What does it represent?

Table 12.14 in the book ↗
x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Answer

The slope is 1.99 (b = 1.99). It means that for every endorsement deal a professional player gets, he gets an average of another $1.99 million in pay each year.

12-25 §12.3 source ↗

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

What does an r value of zero mean?

Table 12.14 in the book ↗
x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Answer

It means that there is no correlation between the data sets.

12-27 §12.3 source ↗

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

When n = 100 and r = -0.89, is there a significant correlation? Explain.

Table 12.14 in the book ↗
x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Answer

Yes, there are enough data points and the value of r is strong enough to show that there is a strong negative correlation between the data sets.

12-29 §12.4 source ↗

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

Answer

Ha: ρ ≠ 0

12-31 §12.5 source ↗

Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as follows:

ŷ = 101.32 + 2.48x where ŷ is in thousands of dollars.

What would you predict the sales to be on day 60?

Answer

$250,120

12-33 §12.5 source ↗


Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,350 acres. The rate at which one person can mow is as follows:

ŷ = 1350 – 1.2x where x is the number of hours and ŷ represents the number of acres left to mow.

How many acres will be left to mow after 20 hours of work?

Answer

1,326 acres

12-35 §12.5 source ↗


Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,350 acres. The rate at which one person can mow is as follows:

ŷ = 1350 – 1.2x where x is the number of hours and ŷ represents the number of acres left to mow.

How many hours will it take to mow all of the lawns? (When is ŷ = 0?)

Answer

1,125 hours, or when x = 1,125

12-37 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

Perform linear regression. What is the linear equation? Round to the nearest whole number.

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

Answers may vary.

12-39 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

Solve.

a. When x = 1985, ŷ = _____

b. When x = 1990, ŷ =_____

c. When x = 1970, ŷ =______ Why doesn’t this answer make sense?

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

a. When x = 1985, ŷ = 25,52

b. When x = 1990, ŷ = 34,275

c. When x = 1970, ŷ = –725 Why doesn’t this answer make sense? The range of x values was 1981 to 2002; the year 1970 is not in this range. The regression equation does not apply, because predicting for the year 1970 is extrapolation, which requires a different process. Also, a negative number does not make sense in this context, where we are predicting flu cases diagnosed.

12-41 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

What does the correlation imply about the relationship between time (years) and the number of diagnosed flu cases reported in the U.S.?

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

Also, the correlation r = 0.4526. If r is compared to the value in the 95% Critical Values of the Sample Correlation Coefficient Table, because r > 0.423, r is significant, and you would think that the line could be used for prediction. But the scatter plot indicates otherwise.

12-43 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

Write the equation: ŷ= ____________

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

y ^ = –3,448,225 + 1750x

12-45 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

Does the line seem to fit the data? Why or why not?

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

There was an increase in flu cases diagnosed until 1993. From 1993 through 2002, the number of flu cases diagnosed declined each year. It is not appropriate to use a linear regression line to fit to the data.

12-47 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

What does the correlation imply about the relationship between time (years) and the number of diagnosed flu cases reported in the U.S.?

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

Since there is no linear association between year and # of flu cases diagnosed, it is not appropriate to calculate a linear correlation coefficient. When there is a linear association and it is appropriate to calculate a correlation, we cannot say that one variable “causes” the other variable.

12-49 §12.5 source ↗

Table 12.15 contains real data for the first two decades of flu cases reporting.

Enter your data into your calculator or computer. The pre-1981 data should not be included. Why is that so?

Write the linear equation, rounding to four decimal places:

Table 12.15 Adults and Adolescents only, United States in the book ↗
Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Answer

We don’t know if the pre-1981 data was collected from a single year. So we don’t have an accurate x value for this figure.

Regression equation: ŷ (#Flu Cases) = –3,448,225 + 1749.777 (year)

Coefficients
Intercept –3,448,225
X Variable 1 1,749.777
Table 12.37
Table 12.37 in the book ↗
Coefficients
Intercept –3,448,225
X Variable 1 1,749.777

12-51 §12.6 source ↗

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

Do there appear to be any outliers?

Figure 12.32 in the book ↗

Answer

Yes, there appears to be an outlier at (6, 58).

12-53 §12.6 source ↗

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

What effect did the potential outlier have on the line of best fit?

Figure 12.32 in the book ↗

Answer

The potential outlier flattened the slope of the line of best fit because it was below the data set. It made the line of best fit less accurate as a predictor for the data.

12-55 §12.6 source ↗

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

The Sum of Squared Errors for a data set of 18 numbers is 49. What is the standard deviation?

Figure 12.32 in the book ↗

Answer

s = 1.75

12-57 §12.1 source ↗

For each of the following situations, state the independent variable and the dependent variable.

a. A study is done to determine if elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared to the age of drivers.

b. A study is done to determine if the weekly grocery bill changes based on the number of family members.

c. Insurance companies base life insurance premiums partially on the age of the applicant.

d. Utility bills vary according to power consumption.

e. A study is done to determine if a higher education reduces the crime rate in a population.

Answer

a. independent variable: age; dependent variable: fatalities

b. independent variable: # of family members; dependent variable: grocery bill

c. independent variable: age of applicant; dependent variable: insurance premium

d. independent variable: power consumption; dependent variable: utility

e. independent variable: higher education (years); dependent variable: crime rates

12-59 §12.2 source ↗

The Gross Domestic Product Purchasing Power Parity is an indication of a country’s currency value compared to another country. Table 12.17 shows the GDP PPP of Cuba as compared to US dollars. Construct a scatter plot of the data.
Table 12.17 in the book ↗
Year Cuba’s PPP Year Cuba’s PPP
1 1,700 8 4,000
2 1,700 9 11,000
4 2,300 10 9,500
5 2,900 11 9,700
6 3,000 12 9,900
7 3,500

Answer

Answers may vary.

12-61 §12.2 source ↗

Does the higher cost of tuition translate into higher-paying jobs? The table lists the top ten colleges based on mid-career salary and the associated yearly tuition costs. Construct a scatter plot of the data.
Table 12.19 in the book ↗
School Mid-Career Salary (in thousands) Yearly Tuition
Princeton 137 57,410
Harvey Mudd 135 62,817
CalTech 127 60,864
US Naval Academy 122 0
West Point 120 0
MIT 118 57,986
Lehigh University 118 59,930
NYU-Poly 117 58,168
Babson College 117 56,576
Stanford 114 56,169

Answer

For graph: Answers may vary. Note that tuition is the independent variable and salary is the dependent variable.

12-63 §12.2 source ↗

If there are 15 data points in a set of data, what is the number of degree of freedom?

Answer

13

12-65 §12.3 source ↗

Explain what it means when a correlation has an r2 of 0.72.

Answer

It means that 72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x).

12-67 §12.5 source ↗

Recently, the annual number of driver deaths per 100,000 for the selected age groups was as follows:

a. For each age group, pick the midpoint of the interval for the x value. (For the 75+ group, use 80.)

b. Using “ages” as the independent variable and “Number of driver deaths per 100,000” as the dependent variable, make a scatter plot of the data.

c. Calculate the least squares (best–fit) line. Put the equation in the form of: ŷ = a + bx

d. Find the correlation coefficient. Is it significant?

e. Predict the number of deaths for ages 40 and 60.

f. Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?

g. What is the slope of the least squares (best-fit) line? Interpret the slope.

Table 12.20 in the book ↗
Age Number of Driver Deaths per 100,000
16–19 38
20–24 36
25–34 24
35–54 20
55–74 18
75+ 28

Answer

a.

Age Number of Driver Deaths per 100,000
16–19 38
20–24 36
25–34 24
35–54 20
55–74 18
75+ 28
Table 12.38

b. Check student’s solution.

c. ŷ = 35.5818045 – 0.19182491x

d. r = –0.57874
For four df and alpha = 0.05, the LinRegTTest gives p-value = 0.2288 so we do not reject the null hypothesis; there is not a significant linear relationship between deaths and age.
Using the table of critical values for the correlation coefficient, with four df, the critical value is 0.811. The correlation coefficient r = –0.57874 is not less than –0.811, so we do not reject the null hypothesis.

e. There is not a linear relationship between the two variables, as evidenced by a p-value greater than 0.05.

Table 12.38 in the book ↗
Age Number of Driver Deaths per 100,000
16–19 38
20–24 36
25–34 24
35–54 20
55–74 18
75+ 28

12-69 §12.5 source ↗

Table 12.22 in the book ↗
Page number Maximum value ($)
4 16
14 19
25 15
32 17
43 19
57 15
72 16
85 15
90 17

The maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition ten, for various pages is given in Table 12.22

a. Decide which variable should be the independent variable and which should be the dependent variable.

b. Draw a scatter plot of the ordered pairs.

c. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx

d. Find the correlation coefficient. Is it significant?

e. Find the estimated maximum values for the restaurants on page ten and on page 70.

f. Does it appear that the restaurants giving the maximum value are placed in the beginning of the “Fine Dining” section? How did you arrive at your answer?

g. Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?

h. Is the least squares line valid for page 200? Why or why not?

i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a. We wonder if the better discounts appear earlier in the book so we select page as X and discount as Y.

b. Check student’s solution.

c. ŷ = 17.21757 – 0.01412x

d. r = – 0.2752
For seven df and alpha = 0.05, using LinRegTTest p-value = 0.4736 so we do not reject; there is a not a significant linear relationship between page and discount.
Using the table of critical values for the correlation coefficient, with seven df, the critical value is 0.666. The correlation coefficient xi = –0.2752 is not less than 0.666 so we do not reject.

e. There is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount.

As the page number increases by one page, the discount decreases by $0.01412

12-71 §12.5 source ↗

Table 12.24 in the book ↗
State # letters in name Year entered the Union Rank for entering the Union Area (square miles)
Alabama 7 1819 22 52,423
Colorado 8 1876 38 104,100
Hawaii 6 1959 50 10,932
Iowa 4 1846 29 56,276
Maryland 8 1788 7 12,407
Missouri 8 1821 24 69,709
New Jersey 9 1787 3 8,722
Ohio 4 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah 4 1896 45 84,904
Wisconsin 9 1848 30 65,499

We are interested in whether or not the number of letters in a state name depends upon the year the state entered the Union.

a. Decide which variable should be the independent variable and which should be the dependent variable.

b. Draw a scatter plot of the data.

c. Does it appear from inspection that there is a relationship between the variables? Why or why not?

d. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx.

e. Find the correlation coefficient. What does it imply about the significance of the relationship?

f. Find the estimated number of letters (to the nearest integer) a state would have if it entered the Union in 1900. Find the estimated number of letters a state would have if it entered the Union in 1940.

g. Does it appear that a line is the best way to fit the data? Why or why not?

h. Use the least-squares line to estimate the number of letters a new state that enters the Union this year would have. Can the least squares line be used to predict it? Why or why not?

Answer

a. Year is the independent or x variable; the number of letters is the dependent or y variable.

b. Check student’s solution.

c. no

d. ŷ = 47.03 – 0.0216x

e. –0.4280 The r-value indicates that there is not a significant correlation between the year the state entered the union and the number of letters in the name.

f. No, the relationship does not appear to be linear; the correlation is not significant.

12-73 §12.6 source ↗

Ornithologists, scientists who study birds, tag sparrow hawks in 13 different colonies to study their population. They gather data for the percent of new sparrow hawks in each colony and the percent of those that have returned from migration.

Percent return:74; 66; 81; 52; 73; 62; 52; 45; 62; 46; 60; 46; 38
Percent new:5; 6; 8; 11; 12; 15; 16; 17; 18; 18; 19; 20; 20

a. Enter the data into your calculator and make a scatter plot.

b. Use your calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a.

c. Explain in words what the slope and y-intercept of the regression line tell us.

d. How well does the regression line fit the data? Explain your response.

e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.

f. An ecologist wants to predict how many birds will join another colony of sparrow hawks to which 70% of the adults from the previous year have returned. What is the prediction?

Answer

a. and b. Check student’s solution.

c. The slope of the regression line is -0.3031 with a y-intercept of 31.93. In context, the y-intercept indicates that when there are no returning sparrow hawks, there will be almost 32% new sparrow hawks, which doesn’t make sense since if there are no returning birds, then the new percentage would have to be 100% (this is an example of why we do not extrapolate). The slope tells us that for each percentage increase in returning birds, the percentage of new birds in the colony decreases by 30.3%.

d. If we examine r2, we see that only 57.52% of the variation in the percent of new birds is explained by the model and the correlation coefficient, r = –.7584 only indicates a somewhat strong correlation between returning and new percentages.

e. The ordered pair (66, 6) generates the largest residual of 6.0. This means that when the observed return percentage is 66%, our observed new percentage, 6%, is almost 6% less than the predicted new value of 11.98%. If we remove this data pair, we see only an adjusted slope of -.2789 and an adjusted intercept of 30.9816. In other words, even though this data generates the largest residual, it is not an outlier, nor is the data pair an influential point.

f. If there are 70% returning birds, we would expect to see y = –.2789(70) + 30.9816 = 0.114 or 11.4% new birds in the colony.

12-75 §12.6 source ↗

The following table consists of one student athlete’s time (in minutes) to swim 2000 yards and the student’s heart rate (beats per minute) after swimming on a random sample of 10 days:

a. Enter the data into your calculator and make a scatter plot.

b. Use your calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a.

c. Explain in words what the slope and y-intercept of the regression line tell us.

d. How well does the regression line fit the data? Explain your response.

e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.

Table 12.27 in the book ↗
Swim Time Heart Rate
34.12 144
35.72 152
34.72 124
34.05 140
34.13 152
35.73 146
36.17 128
35.57 136
35.37 144
35.57 148

Answer

a. Check student’s solution.

b. Check student’s solution.

c. We have a slope of –1.4946 with a y-intercept of 193.88. The slope, in context, indicates that for each additional minute added to the swim time, the heart rate will decrease by 1.5 beats per minute. If the student is not swimming at all, the y-intercept indicates that his heart rate will be 193.88 beats per minute. While the slope has meaning (the longer it takes to swim 2,000 meters, the less effort the heart puts out), the y-intercept does not make sense. If the athlete is not swimming (resting), then his heart rate should be very low.

d. Since only 1.5% of the heart rate variation is explained by this regression equation, we must conclude that this association is not explained with a linear relationship.

e. The point (34.72, 124) generates the largest residual of –11.82. This means that our observed heart rate is almost 12 beats less than our predicted rate of 136 beats per minute. When this point is removed, the slope becomes –2.953 with the y-intercept changing to 247.1616. While the linear association is still very weak, we see that the removed data pair can be considered an influential point in the sense that the y-intercept becomes more meaningful.

12-77 §12.6 source ↗

Using the data to determine the linear-regression line equation with the outliers removed. Is there a linear correlation for the data set with outliers removed? Justify your answer.
Table 12.29 in the book ↗
School Mid-Career Salary (in thousands) Yearly Tuition
Princeton 137 57,410
Harvey Mudd 135 62,817
CalTech 127 60,864
US Naval Academy 122 0
West Point 120 0
MIT 118 57,986
Lehigh University 118 59,930
NYU-Poly 117 58,168
Babson College 117 56,576
Stanford 114 56,169

Answer

If we remove the two service academies (the tuition is $0.00), we construct a new regression equation of y = 0.002x + 0.309 with a correlation coefficient of 0.533 and a coefficient of determination of 0.284. There is not enough evidence to state that there is a strong linear correlation between tuition costs and future salaries, even when the service academies are removed.

12-79 §12.BIT source ↗

Table 12.31 in the book ↗
Year Percent of workers paid hourly rates
1 61.2
2 60.7
3 61.3
4 61.3
5 61.8
6 61.7
7 61.8
8 62.0
9 62.7
12 62.8
14 62.9

The percent of women who are wage and salary workers who are paid hourly rates is given in Table 12.31 for a 14-year period.

a. Using “year” as the independent variable and “percent” as the dependent variable, draw a scatter plot of the data.

b. Does it appear from inspection that there is a relationship between the variables? Why or why not?

c. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx

d. Find the correlation coefficient. Is it significant?

e. Find the estimated percents for year 13 and year 10.

f. Based on the data, is there a linear relationship between the year and the percent of female wage and salary earners who are paid hourly rates?

g. Are there any outliers in the data?

h. What is the estimated percent for the year 72? Does the least-squares line give an accurate estimate for that year? Explain why or why not.

i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a. Check student's solution.

b. yes

c. ŷ = −266.8863+0.1656x

d. 0.9448; Yes

e. 62.8233; 62.3265

f. yes

g. no; (Year 9, 62.7)

h. 72.5937; no

i. slope = 0.1656.
As the year increases by one, the percent of workers paid hourly rates tends to increase by 0.1656.

12-81 §12.BIT source ↗

Table 12.32 in the book ↗
Size (ounces) Cost ($) Cost per ounce
16 3.99
32 4.99
64 5.99
200 10.99


Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in Table 12.32.

a. Complete Table 12.32 for the cost per ounce of the different sizes.

b. Using “size” as the independent variable and “cost per ounce” as the dependent variable, draw a scatter plot of the data.

c. Does it appear from inspection that there is a relationship between the variables? Why or why not?

d. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx

e. Find the correlation coefficient. Is it significant?

f. If the laundry detergent were sold in a 40-ounce size, find the estimated cost per ounce.

g. If the laundry detergent were sold in a 90-ounce size, find the estimated cost per ounce.

h. Does it appear that a line is the best way to fit the data? Why or why not?

i. Are there any outliers in the the data?

j. Is the least-squares line valid for predicting what a 300-ounce size of the laundry detergent would cost per ounce? Why or why not?

k. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a.

Size (ounces) Cost ($) cents/oz
16 3.99 24.94
32 4.99 15.59
64 5.99 9.36
200 10.99 5.50
Table 12.39

b. Check student’s solution.

c. There is a linear relationship for the sizes 16 through 64, but that linear trend does not continue to the 200-oz size.

d. ŷ = 20.2368 – 0.0819x

e. r = –0.8086

f. 40-oz: 16.96 cents/oz

g. 90-oz: 12.87 cents/oz

h. The relationship is not linear; the least squares line is not appropriate.

i. no outliers

j. No, you would be extrapolating. The 300-oz size is outside the range of x.

k. slope = –0.08194; for each additional ounce in size, the cost per ounce decreases by 0.082 cents.

Table 12.39 in the book ↗
Size (ounces) Cost ($) cents/oz
16 3.99 24.94
32 4.99 15.59
64 5.99 9.36
200 10.99 5.50

12-83 §12.BIT source ↗

Table 12.32 in the book ↗
Size (ounces) Cost ($) Cost per ounce
16 3.99
32 4.99
64 5.99
200 10.99
Table 12.34 in the book ↗
Hard Disk Drive Capacity (in terabytes) Sale Price ($)
9 147
20 197
27 297
31 447
35 1177
40 2177
60 2497


Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in Table 12.32.

The following are advertised sale prices of hard disk drives on Amazon’s website for various capacity hard drives.

a. Decide which variable should be the independent variable and which should be the dependent variable.

b. Draw a scatter plot of the data.

c. Does it appear from inspection that there is a relationship between the variables? Why or why not?

d. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx

e. Find the correlation coefficient. Is it significant?

f. Find the estimated sale price for a 32 terabyte hard drive. Find the cost for a 50 terabyte hard drive.

g. Does it appear that a line is the best way to fit the data? Why or why not?

h. Are there any outliers in the data?

i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a. Size is x, the independent variable, price is y, the dependent variable.

b. Answers may vary.

c. The relationship does not appear to be linear.

d. ŷ = –745.252 + 54.75569x

e. r = 0.8944, yes it is significant

f. 32-terabyte: $1006.93, 50-terabyte: $1992.53

g. No, the relationship does not appear to be linear. However, r is significant.

h. no, the 60-terabyte hard drive

i. For each additional terabyte, the price increases by $54.76

12-85 §12.BIT source ↗

Table 12.32 in the book ↗
Size (ounces) Cost ($) Cost per ounce
16 3.99
32 4.99
64 5.99
200 10.99
Table 12.36 in the book ↗
State # letters in name Year entered the Union Ranks for entering the Union Area (square miles)
Alabama 7 1819 22 52,423
Colorado 8 1876 38 104,100
Hawaii 6 1959 50 10,932
Iowa 4 1846 29 56,276
Maryland 8 1788 7 12,407
Missouri 8 1821 24 69,709
New Jersey 9 1787 3 8,722
Ohio 4 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah 4 1896 45 84,904
Wisconsin 9 1848 30 65,499


Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in Table 12.32.

We are interested in whether there is a relationship between the ranking of a state and the area of the state.

a. What are the independent and dependent variables?

b. What do you think the scatter plot will look like? Make a scatter plot of the data.

c. Does it appear from inspection that there is a relationship between the variables? Why or why not?

d. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx

e. Find the correlation coefficient. What does it imply about the significance of the relationship?

f. Find the estimated areas for Alabama and for Colorado. Are they close to the actual areas?

g. Use the two points in part f to plot the least-squares line on your graph from part b.

h. Does it appear that a line is the best way to fit the data? Why or why not?

i. Are there any outliers?

j. Use the least squares line to estimate the area of a new state that enters the Union. Can the least-squares line be used to predict it? Why or why not?

k. Delete “Hawaii” and substitute “Alaska” for it. Alaska is the forty-ninth, state with an area of 656,424 square miles.

l. Calculate the new least-squares line.

m. Find the estimated area for Alabama. Is it closer to the actual area with this new least-squares line or with the previous one that included Hawaii? Why do you think that’s the case?

n. Do you think that, in general, newer states are larger than the original states?

Answer

a. Let rank be the independent variable and area be the dependent variable.

b. Answers may vary.

c. There appears to be a linear relationship, with one outlier.

d. ŷ (area) = 24177.06 + 1010.478x

e. r = 0.50047, r is not significant so there is no relationship between the variables.

f. Alabama: 46407.576 Colorado: 62575.224

g. Alabama estimate is closer than Colorado estimate.

h. If the outlier is removed, there is a linear relationship.

i. There is one outlier (Hawaii).

j. rank 51: 75711.4; no

k.

Alabama 7 1819 22 52,423
Colorado 8 1876 38 104,100
Hawaii 6 1959 50 10,932
Iowa 4 1846 29 56,276
Maryland 8 1788 7 12,407
Missouri 8 1821 24 69,709
New Jersey 9 1787 3 8,722
Ohio 4 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah 4 1896 45 84,904
Wisconsin 9 1848 30 65,499
Table 12.40

l. ŷ = –87065.3 + 7828.532x

m. Alabama: 85,162.404; the prior estimate was closer. Alaska is an outlier.

n. yes, with the exception of Hawaii

Table 12.40 in the book ↗
Alabama 7 1819 22 52,423
Colorado 8 1876 38 104,100
Hawaii 6 1959 50 10,932
Iowa 4 1846 29 56,276
Maryland 8 1788 7 12,407
Missouri 8 1821 24 69,709
New Jersey 9 1787 3 8,722
Ohio 4 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah 4 1896 45 84,904
Wisconsin 9 1848 30 65,499

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Problems and answers are adapted from Introductory Statistics 2e by Barbara Illowsky and Susan Dean, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.