Introductory Statistics

Barbara Illowsky, Susan Dean

Chapter 12

Linear Regression and Correlation

Educators

LE
MP
JB
+ 3 more educators

Problem 1

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?

LE
Lucas E.
Numerade Educator

Problem 2

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.

Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

LE
Lucas E.
Numerade Educator

Problem 3

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.

LE
Lucas E.
Numerade Educator

Problem 4

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

Find the equation that expresses the total fee in terms of the number of days the payment is late.

LE
Lucas E.
Numerade Educator

Problem 5

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

Graph the equation from Exercise 12.4.

LE
Lucas E.
Numerade Educator

Problem 6

Is the equation $y=10+5 x-3 x^{2}$ linear? Why or why not?

LE
Lucas E.
Numerade Educator

Problem 7

Which of the following equations are linear?
a. $y=6 x+8$
b. $y+7=3 x$
c. $y-x=8 x^{2}$
d. $4 y=8$

LE
Lucas E.
Numerade Educator

Problem 8

Does the graph show a linear equation? Why or why not?

Figure 12.25 (Cant copy)

Table 12.12 contains real data for the first two decades of AIDS reporting.

$$\begin{array}{|l|l|}\hline \text { Year } & {\# \text { AlDS cases diagnosed }} & {\# \text { AlDS deaths }} \\ \hline \text { Pre-1981 } & {91} & {29} \\ \hline 1981 & {319} & {121} \\ \hline 1982 & {1,170} & {453}\\ \hline 1983 & {3,076} & {1,482} \\ \hline 1984 & {6,240} & {3,466} \\ \hline 1985 & {11,776} & {6,878} \\ \hline 1986 & {19,032} & {11,987} \\ \hline 1987 & {28,564} & {16,162} \\ \hline 1988 & {35,447} & {20,868} \\ \hline 1989 & {42,674} & {27,591} \\ \hline 1990 & {48,634} & {31,335} \\ \hline 1991 & {59,660} & { 36,560} \\ \hline 1992 & {78,530} & {41,055} \\ \hline 1993 & {78,834} & {44,730} \\ \hline 1994 & {71,874} & {49,095} \\ \hline 1995 & {68,505} & {49, 456} \\ \hline 1996 & {59,347} & {38,510} \\ \hline 1997 & {47,149} & {20,736} \\ \hline 1998 & {38,393} & {19,005} \\ \hline 1999 & {25,174} & {18,454} \\ \hline 2000 & {25,522} & {17, 347} \\ \hline 2001 & {25,643} & {17,402} \\ \hline 2002 & {26,464} & {16,371} \\ \hline \text { Total } & {802,118} & {489,093} \\ \hline \end{array}$$
Table 12.12 Adults and Adolescents only, United States

LE
Lucas E.
Numerade Educator

Problem 9

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

LE
Lucas E.
Numerade Educator

Problem 10

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 $y=50+100 x$ .

What are the independent and dependent variables?

JB
Josephine B.
Numerade Educator

Problem 11

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 $y=50+100 x$ .

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

William S.
Numerade Educator

Problem 12

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 $y=12,000 x$ .

What are the independent and dependent variables?

William S.
Numerade Educator

Problem 13

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 $y=12,000 x$ .

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

Nick K.
Numerade Educator

Problem 14

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 $y=12,000 x$ .

What is the $y$ -intercept? Interpret its meaning.

William S.
Numerade Educator

Problem 15

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 $y=15-1.5 x$ 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.

William S.
Numerade Educator

Problem 16

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 $y=15-1.5 x$ where $x$ is the number of hours passed in an eight-hour day of trading.

If you owned this stock, would you want a positive or negative slope? Why?

William S.
Numerade Educator

Problem 17

Does the scatter plot appear linear? Strong or weak? Positive or negative?

William S.
Numerade Educator

Problem 18

Does the scatter plot appear linear? Strong or weak? Positive or negative?

William S.
Numerade Educator

Problem 19

Does the scatter plot appear linear? Strong or weak? Positive or negative?

William S.
Numerade Educator

Problem 20

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).

$$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\ \hline\end{array}$$
Table 12.13

Draw a scatter plot of the data.

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Problem 21

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).

$$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\ \hline\end{array}$$
Table 12.13

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

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Problem 22

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).

$$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\ \hline\end{array}$$
Table 12.13

Draw the line of best fit on the scatter plot.

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Problem 23

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).

$$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\ \hline\end{array}$$
Table 12.13

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

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Problem 24

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).

$$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\ \hline\end{array}$$
Table 12.13

What is the $y$-intercept of the line of best fit? What does it represent?

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Problem 25

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).

$$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\ \hline\end{array}$$
Table 12.13

What does an $r$ value of zero mean?

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Problem 26

When $n=2$ and $r=1,$ are the data significant? Explain.

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Problem 27

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

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Problem 28

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

Trang H.
Numerade Educator

Problem 29

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

Trang H.
Numerade Educator

Problem 30

If the level of significance is 0.05 and the $p$ -value is $0.04,$ what conclusion can you draw?

Trang H.
Numerade Educator

Problem 31

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: $\hat{y}=101.32+2.48 x$ where $\hat{y}$ is in thousands of dollars.

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

Donald A.
Numerade Educator

Problem 32

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: $\hat{y}=101.32+2.48 x$ where $\hat{y}$ is in thousands of dollars.

What would you predict the sales to be on day 90?

Donald A.
Numerade Educator

Problem 33

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 1,345 acres. The rate at which one person can mow is as follows:
$\hat{y}=1350-1.2 x$ where $x$ is the number of hours and $\hat{y}$ represents the number of acres left to mow.

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

Donald A.
Numerade Educator

Problem 34

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 1,345 acres. The rate at which one person can mow is as follows:
$\hat{y}=1350-1.2 x$ where $x$ is the number of hours and $\hat{y}$ represents the number of acres left to mow.

How many acres will be left to mow after 100 hours of work?

Donald A.
Numerade Educator

Problem 35

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 1,345 acres. The rate at which one person can mow is as follows:
$\hat{y}=1350-1.2 x$ where $x$ is the number of hours and $\hat{y}$ represents the number of acres left to mow.

How many hours will it take to mow all of the lawns? (When is $\hat{y}=0 ?$

Donald A.
Numerade Educator

Problem 36

Table 12.14 contains real data for the first two decades of AIDS reporting.
(Table 12.14 Can't Copy)

Graph “year” versus “# AIDS cases diagnosed” (plot the scatter plot). Do not include pre-1981 data.

Donald A.
Numerade Educator

Problem 37

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

Donald A.
Numerade Educator

Problem 38

Write the equations:
a. Linear equation: __________
b. $a$ = ________
c. $b$ = ________
d. $r$ = ________
e. $n$ = ________

Donald A.
Numerade Educator

Problem 39

Solve.
a. When x = 1985, $\hat{y}$ = _____
b. When x = 1990, $\hat{y}$ =_____
c. When x = 1970, $\hat{y}$ =______ Why doesn’t this answer make sense?

Donald A.
Numerade Educator

Problem 40

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

Donald A.
Numerade Educator

Problem 41

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

Donald A.
Numerade Educator

Problem 42

Plot the two given points on the following graph. Then, connect the two points to form the regression line.

(Graph can't copy)

Donald A.
Numerade Educator

Problem 43

Obtain the graph on your calculator or computer.

Write the equation: $\hat{y}=$ ____________

Donald A.
Numerade Educator

Problem 44

Obtain the graph on your calculator or computer.

Hand draw a smooth curve on the graph that shows the flow of the data.

Donald A.
Numerade Educator

Problem 45

Obtain the graph on your calculator or computer.

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

Donald A.
Numerade Educator

Problem 46

Obtain the graph on your calculator or computer.

Do you think a linear fit is best? Why or why not?

Donald A.
Numerade Educator

Problem 47

Obtain the graph on your calculator or computer.

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

Donald A.
Numerade Educator

Problem 48

Obtain the graph on your calculator or computer.

Graph “year” vs. “# AIDS cases diagnosed.” Do not include pre-1981. Label both axes with words. Scale both axes.

Donald A.
Numerade Educator

Problem 49

Obtain the graph on your calculator or computer.

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

Donald A.
Numerade Educator

Problem 50

Write the linear equation, rounding to four decimal places:
Calculate the following:
a. a = _____
b. b = _____
c. correlation = _____
d. n = _____

Donald A.
Numerade Educator

Problem 51

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.

(Graph Cant copy)

Do there appear to be any outliers?

Donald A.
Numerade Educator

Problem 52

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.

(Graph Cant copy)

A point is removed, and the line of best fit is recalculated. The new correlation coefficient is 0.98. Does the point appear to have been an outlier? Why?

Donald A.
Numerade Educator

Problem 53

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.

(Graph Cant copy)

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

Donald A.
Numerade Educator

Problem 54

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.

(Graph Cant copy)

Are you more or less confident in the predictive ability of the new line of best fit?

Donald A.
Numerade Educator

Problem 55

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

Donald A.
Numerade Educator

Problem 56

The Standard Deviation for the Sum of Squared Errors for a data set is 9.8. What is the cutoff for the vertical distance that a point can be from the line of best fit to be considered an outlier?

Donald A.
Numerade Educator

Problem 57

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.

Ashka D.
Numerade Educator

Problem 58

Piece-rate systems are widely debated incentive payment plans. In a recent study of loan officer effectiveness, the following piece-rate system was examined:

( Table 12.15 Can't Copy)

If a loan officer makes 95% of his or her goal, write the linear function that applies based on the incentive plan table. In context, explain the y-intercept and slope.

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Problem 59

The Gross Domestic Product Purchasing Power Parity is an indication of a country’s currency value compared to another country. Table 12.16 shows the GDP PPP of Cuba as compared to US dollars. Construct a scatter plot of the data.

$$\begin{array}{|c|c|}\hline \text { Year } & {\text { Cuba's PPP }} & {\text { Year }} & {\text { Cuba's PPP }} \\ \hline 1999 & {1,700} & {2006} & {4,000} \\ \hline 2000 & {1,700} & {2007} & {11,000} \\ \hline 2002 & {2,300} & {2008} & {9,500} \\ \hline 2003 & {2,900} & {2009} & {9,900} \\ \hline 2004 & {3,000} & {2010} & {9,900} \\ \hline 2005 & {3,500} & {} \\ \hline\end{array}$$
Table 12.16

William S.
Numerade Educator

Problem 60

The following table shows the poverty rates and cell phone usage in the United States. Construct a scatter plot of the data

$$\begin{array}{|l|l|}\hline \text { Year } & {\text { Poverty Rate }} & {\text { Cellular Usage per Capita }} \\ \hline 2003 & {12.7} & {54.67} \\ \hline 2005 & {12.6} & {74.19} \\ \hline 2007 & {12} & {84.86} \\ \hline 2009 & {12} & {90.82} \\ \hline \end{array}$$
Table 12.17

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Problem 61

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 12.18 Can't Copy)

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Problem 62

If the level of significance is 0.05 and the $p$-value is 0.06, what conclusion can you draw?

William S.
Numerade Educator

Problem 63

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

William S.
Numerade Educator

Problem 64

What is the process through which we can calculate a line that goes through a scatter plot with a linear pattern?

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Problem 65

Explain what it means when a correlation has an $r^{2}$ of 0.72.

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Problem 66

Can a coefficient of determination be negative? Why or why not?

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Problem 67

If the level of significance is 0.05 and the $p$ -value is $0.06,$ what conclusion can you draw?

Trang H.
Numerade Educator

Problem 68

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

Trang H.
Numerade Educator

Problem 69

Recently, the annual number of driver deaths per 100,000 for the selected age groups was as follows:
(TABLE CAN'T COPY)
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: $\hat{y}=a+b x$
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.

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Problem 70

Table 12.20 shows the life expectancy for an individual born in the United States in certain years.
$$\begin{array}{|c|c|}\hline \text { Year of Birth } & {\text { Life Expectancy }} \\ \hline 1930 & {59.7} \\ \hline 1940 & {62.9} \\ \hline 1950 & {70.2} \\ \hline 1950 & {71.5} \\ \hline 1987 & {75} \\ \hline 1987 & {75.7} \\ \hline 2010 & {78.7} \\ \hline\end{array}$$
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: $\hat{y}=a+b x$
d. Find the correlation coefficient. Is it significant?
e. Find the estimated life expectancy for an indidual born in 1950 and for one born in 1982
f. Why aren't the answers to part e the same as the values in Table 12.20 that correspond to those years?
g. Use the two points in part e to plot the least squares line on your graph from part b.
h. Based on the data, is there a linear relationship between the year of birth and life expectancy?
i. Are there any outliers in the data?
j. Using the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not.
k. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Donald A.
Numerade Educator

Problem 71

The maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition ten, for various pages is given in Table 12.21
$$\begin{array}{|c|c|}\hline \text { Page number } & {\text { Maximum value (s) }} \\ \hline 4 & {16} \\ \hline 14 & {19} \\ \hline 25 & {19} \\ \hline 25 & {17} \\ \hline 43 & {17} \\ \hline 42 & {15} \\ \hline 72 & {15} \\ \hline 85 & {17} \\ \hline 90 & {17} \\ \hline\end{array}$$
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: $\hat{y}=a+b x$
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.

Donald A.
Numerade Educator

Problem 72

Table 12.22 gives the gold medal times for every other Summer Olympics for the women’s 100-meter freestyle (swimming).
$$
\begin{array}{|c|c|}\hline \text { Year } & {\text { Time (seconds) }} \\ \hline 1912 & {82.2} \\ \hline 1924 & {72.4} \\ \hline 1932 & {66.8} \\ \hline 1952 & {66.8} \\ \hline 1960 & {61.2} \\ \hline 1968 & {60.0} \\ \hline 1968 & {55.65} \\ \hline\end{array}
$$
$$
\begin{array}{|c|c|}\hline \text { Year } & {\text { Time (seconds) }} \\ \hline 1984 & {55.92} \\ \hline 1992 & {54.64} \\ \hline 2000 & {53.8} \\ \hline 2008 & {53.1} \\ \hline\end{array}
$$
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: $\hat{y}=a+b x .$
e. Find the correlation coefficient. Is the decrease in times significant?
f. Find the estimated gold medal time for 1932. Find the estimated time for 1984.
g. Why are the answers from part f different from the chart values?
h. Does it appear that a line is the best way to fit the data? Why or why not?
i. Use the least-squares line to estimate the gold medal time for the next Summer Olympics. Do you think that your answer is reasonable? Why or why not?

Donald A.
Numerade Educator

Problem 73

(TABLE CAN'T COPY)
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: $\hat{y}=a+b x$
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?

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Problem 74

The height (sidewalk to roof) of notable tall buildings in America is compared to the number of stories of the building (beginning at street level).
(TABLE CAN'T COPY)
a. Using “stories” as the independent variable and “height” as the dependent variable, make a scatter plot of the data.
b. Does it appear from inspection that there is a relationship between the variables?
c. Calculate the least squares line. Put the equation in the form of: $\hat{y}=a+b x$
d. Find the correlation coefficient. Is it significant?
e. Find the estimated heights for 32 stories and for 94 stories.
f. Based on the data in Table 12.24, is there a linear relationship between the number of stories in tall buildings and the height of the buildings?
g. Are there any outliers in the data? If so, which point(s)?
h. What is the estimated height of a building with six stories? Does the least squares line give an accurate estimate of height? Explain why or why not.
i. Based on the least squares line, adding an extra story is predicted to add about how many feet to a building?
j. What is the slope of the least squares (best-fit) line? Interpret the slope.

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Problem 75

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?

Donald A.
Numerade Educator

Problem 76

The following table shows data on average per capita wine consumption and heart disease rate in a random sample of 10 countries.
(TABLE CAN'T COPY)
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. Do the data provide convincing evidence that there is a linear relationship between the amount of alcohol consumed and the heart disease death rate? Carry out an appropriate test at a significance level of 0.05 to help answer this question.

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Problem 77

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:
(TABLE CAN'T COPY)
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.

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Problem 78

A researcher is investigating whether non-white minorities commit a disproportionate number of homicides. He uses demographic data from Detroit, MI to compare homicide rates and the number of the population that are white males.
(TABLE CAN'T COPY)
a. Use your calculator to construct a scatter plot of the data. What should the independent variable be? Why?
b. Use your calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot.
c. Discuss what the following mean in context.
$$\begin{array}{l}{\text { i. The slope of the regression equation }} \\ {\text { ii. The y-intercept of the regression equation }} \\ {\text { iil. The correlation } \mathrm{r}} \\ {\text { iv. The coefficient of determination r2. }}\end{array}$$
d. Do the data provide convincing evidence that there is a linear relationship between the number of white males in the population and the homicide rate? Carry out an appropriate test at a significance level of 0.05 to help answer this question.

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Problem 79

(TABLE CAN'T COPY)
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.

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Problem 80

The average number of people in a family that received welfare for various years is given in Table 12.29.
(TABLE CAN'T COPY)
a. Using "year" as the independent variable and "welfare family size" as the dependent variable, draw a scatter plot of the data.
b. Calculate. the least-squares line. Put the equation in the form of: $\hat{y}=a+b x$
c. Find the correlation coefficient. Is it significant?
d. Pick two years between 199 and 1991 and find the estimated welfare family sizes.
e. Based on the data in Table 12.29, is there a linear relationship between the year and the average number of people in a welfare family?
f. Using the least-squares line, estimate the welfare family sizes for 1960 and 1995. Does the least-squares line give an accurate estimate for those years? Explain why or why not.
g. Are there any outliers in the data?
h. What is the estimated average welfare family size for 1986? 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.

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Problem 81

The percent of female wage and salary workers who are paid hourly rates is given in Table 12.30 for the years 1979 to 1992.
$$\begin{array}{|l|l|}\hline \text { Year } & {\text { Percent of workers paid hourly rates }} \\ \hline 1979 & {61.2} \\ \hline 1980 & {60.7} \\ \hline 1981 & {61.3} \\ \hline 1982 & {61.3} \\ \hline\end{array}$$
$$\begin{array}{|l|l|}\hline \text { Year } & {\text { Percent of workers paid hourly rates }} \\ \hline 1983 & {61.8} \\ \hline 1984 & {61.7} \\ \hline 1985 & {61.8} \\ \hline 1986 & {61.8} \\ \hline 1987 & {62.7} \\ \hline 1987 & {62.8} \\ \hline 1992 & {62.9} \\ \hline\end{array}$$
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: $\hat{y}=a+b x$
d. Find the correlation coefficient. Is it significant?
e. Find the estimated percents for 1991 and 1988.
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 2050? 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.

Donald A.
Numerade Educator

Problem 82

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 12.31.
$$\begin{array}{|c|c|}\hline \text { Size (ounces) } & {\text { cost }(\mathrm{s})} & {\text { cost per ounce }} \\ \hline 16 & {3.99} \\ \hline 32 & {4.99} \\ \hline 64 & {5.99} \\ \hline 600 & {10.99} \\ \hline\end{array}
$$
a. Using “size” as the independent variable and “cost” as the dependent variable, draw a scatter plot.
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: $\hat{y}=a+b x$
d. Find the correlation coefficient. Is it significant?
e. If the laundry detergent were sold in a 40-ounce size, find the estimated cost.
f. If the laundry detergent were sold in a 90-ounce size, find the estimated cost.
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 given data?
i. Is the least-squares line valid for predicting what a 300-ounce size of the laundry detergent would you cost? Why or why not?
j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Donald A.
Numerade Educator

Problem 83

a. Complete Table 12.31 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: $\hat{y}=a+b x$
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.

Donald A.
Numerade Educator

Problem 84

According to a flyer by a Prudential Insurance Company representative, the costs of approximate probate fees and taxes for selected net taxable estates are as follows:
(TABLE CAN'T COPY)
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: $\hat{y}=a+b x$
e. Find the correlation coefficient. Is it significant?
f. Find the estimated total cost for a next taxable estate of $\$ 1,000,000 .$ Find the cost for $\$ 2,500,000$ .
g. Does it appear that a line is the best way to fir the data? Why or why not?
h. Are there any outliers in the data?
i. Based on these results, what would be the probate fees and taxes for an estate that does not have any assets?
j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

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Problem 85

The following are advertised sale prices of color televisions at Anderson’s.
$$\begin{array}{|c|c|}\hline \text { Size (inches) } & {\text { Sale Price (s) }} \\ \hline 9 & {147} \\ \hline 9 & {147} \\ \hline 20 & {197} \\ \hline 27 & {297} \\ \hline 35 & {447} \\ \hline 40 & {2177} \\ \hline 60 & {2497} \\ \hline\end{array}$$
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: $\hat{y}=a+b x$
e. Find the correlation coefficient. Is it significant?
f. Find the estimated sale price for a 32 inch television. Find the cost for a 50 inch television.
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.

Donald A.
Numerade Educator

Problem 86

Table 12.34 shows the average heights for American boy s in 1990.
$$\begin{array}{|c|c|}\hline \text { Age (years) } & {\text { Height (cm) }} \\ \hline \text { birth } & {50.8} \\ \hline 2 & {83.8} \\ \hline 3 & {91.4} \\ \hline 5 & {106.6} \\ \hline 7 & {119.3} \\ \hline 10 & {137.1} \\ \hline 14 & {157.5} \\ \hline\end{array}$$
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: $\hat{y}=a+b x$
e. Find the correlation coefficient. Is it significant?
f. Find the estimated average height for a one-year-old. Find the estimated average height for an eleven-year-old.
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. Use the least squares line to estimate the average height for a sixty-two-year-old man. Do you think that your answer is reasonable? Why or why not?
j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Donald A.
Numerade Educator

Problem 87

(TABLE CAN'T COPY)
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: $\hat{y}=a+b x$
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?

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