Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

Chapter 25

Inferences for Regression - all with Video Answers

Educators

Chapter Questions

Problem 1

Graduation rates 2013 A prestigious college is interested in factors that might be associated with better graduation rates. The administrators wonder whether there is a relationship between acceptance rates and graduation rates. Before proceeding with their regression inference, what conditions and assumptions must be satisfied? Are they? (Source: US News and World Report National University Rankings October $2013 .$ http://colleges .usnews.rankingsandreviews.com/best-colleges/rankings /national-universities/data/spp' $\mathrm{p} \% 2 \mathrm{~B} 50 / \mathrm{page}+1)$
Response variable is Grad\% R-squared $=75.2 \%$ with $50-2=48$ $s=0.0254$
degrees of freedom
$\begin{array}{lclll}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 0.975175 & 0.0074 & 131 & \leq 0.0001 \\ \text { Acceptance \% } & -0.284895 & 0.0236 & -12.1 & \leq 0.0001\end{array}$

James Kiss

James Kiss

Numerade Educator

Problem 2

Shoot to score A college hockey coach collected data from the $2010-2011$ National Hockey League season. He hopes to convince his players that the number of shots taken has an affect on the number of goals scored. The coach performed a preliminary analysis, using the scoring statistics from 293 offensive players who play professional hockey. He predicts Goals from number of Shots (taken for the season). Discuss each of the conditions and assumptions required for him to proceed with the regression analysis. (www.nhl.com)
Dependent variable is Goals R-squared $=63.4 \%$ $s=5.13$ with $293-2=291$ degrees of freedom
$\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -1.77095 & 0.9087 & -1.9488 & 0.052 \\ \text { Shots } & 0.11938 & 0.0053 & 22.460 & <0.0001\end{array}$

James Kiss

Numerade Educator

Problem 3

Graduation rates $2013,$ part II Using the regression output in Exercise 1 , identify the error standard deviation and explain what it means in the context of the problem.

James Kiss

Numerade Educator

Problem 4

Shoot to score another one Using the regression output from Exercise 2 , identify the error standard deviation and explain its meaning with a sentence in context.

James Kiss

Numerade Educator

Problem 5

Graduation rates $2013,$ part III Continuing with the regression of Exercise $1,$ write a sentence that explains the meaning of the standard error of the slope of the regression line, $S E\left(b_{1}\right)=0.0240$.

James Kiss

Numerade Educator

Problem 6

Shoot to score, hat trick Returning to the results of Exercise $2,$ write a sentence to explain the meaning of the standard error of the slope of the regression line, $S E\left(b_{1}\right)=0.0053$

Robin Corrigan

Numerade Educator

Problem 7

Graduation 2013 , part IV The college administrators in Exercise 1 tested the hypotheses $\mathrm{H}_{0}: \beta_{1}=0$ vs. $\mathrm{H}_{\mathrm{A}}$ :
$\beta_{1} \neq 0$ and rejected the null hypothesis because the P-value was less than 0.0001 . What can they conclude about the relationship between admission rates and graduation rates?

James Kiss

Numerade Educator

Problem 8

Shoot to score, number four What can the hockey coach in Exercise 2 conclude about shooting and scoring goals from the fact that the P-value $<0.0001$ for the slope of the regression line? Write a sentence in context.

James Kiss

Numerade Educator

Problem 9

Graduation $2013,$ part $\mathrm{V}$ The college administrators in Exercise 1 constructed a $95 \%$ confidence interval for the slope of their regression line. Interpret the meaning of their interval (-0.332,-0.238) within the context of the problem.

James Kiss

Numerade Educator

Problem 10

Shoot to score, overtime The coach in Exercise 2 found a $95 \%$ confidence interval for the slope of his regression line. Recall that he is trying to predict total goals scored based on shots taken. Interpret with a sentence the meaning of the interval $0.12 \pm 0.01$.

James Kiss

Numerade Educator

Problem 11

Graduation $2013,$ part VI The college described in Exercise 1 admits about $33 \%$ of applicants each year. Which will be larger, the standard error of the predicted 6-Year Graduation Rate of this college, or the standard error of the mean 6 -Year Graduation Rate of all colleges admitting $33 \%$ of their applicants? Explain briefly.

James Kiss

Numerade Educator

Problem 12

Shoot to score, double overtime One of the coach's best players has been in a slump, and the coach is trying to convince him to take more shots. Last year, the player took 104 shots. Would the standard error of the predicted number of goals he shot be larger or smaller than the standard error of the mean predicted number of all players who took 104 shots? Explain briefly.

James Kiss

Numerade Educator

Problem 13

Graduation 2013 , part VII As in Exercise 11 , the college described in Exercise 1 admits about $33 \%$ of applicants each year. Using their regression model, the administrators calculated a $95 \%$ confidence interval for the mean 6 -Year Graduation Rate for all top colleges who admit $33 \%$ of students. The interval was $(87.35 \%, 88.88 \%) .$ Interpret the meaning of this interval with a sentence.

James Kiss

Numerade Educator

Problem 14

Shoot to score, triple overtime As in Exercise $12,$ one of the coach's best players has been in a slump, and the coach is trying to convince him to take more shots. Last year, the player took 104 shots. The coach calculated a $95 \%$ confidence interval for the mean number of goals a player would score if 104 shots were taken. The interval turned out to be 9.8 to 11.5 goals. Summarize the interval with a sentence in context.

James Kiss

Numerade Educator

Problem 15

Graduation 2013 , part VIII The administrators from Exercise 1 also determined a $95 \%$ prediction interval for the graduation rate of a school admitting $33 \%$ of applicants to be $(82.96 \%, 93.28 \%) .$ Carefully explain what the $95 \%$ indicates for this interval.

James Kiss

Numerade Educator

Problem 16

Shoot to score, again The coach, still working with his superstar from Exercise $12,$ went on to suggest the player shoot more. Hopeful for 150 shots for the season, he calculated a $95 \%$ prediction interval for the number of goals this player might score. If the prediction interval for goals scored was (6.03,24.25) , interpret the meaning of this interval with a sentence in context.

James Kiss

Numerade Educator

Problem 17

Graduation 2013 party Would it be wise for the administrators at the college described in Exercise 1 to conclude that lowering their admission rate (i.e., making it harder to get in) would help to increase their graduation rates? Why or why not?

James Kiss

Numerade Educator

Problem 18

Shoot to score, shootout The coach from Exercise 2 called a team meeting to summarize the results from his study. Would it be a good strategy to tell the players that all they need to do is to shoot more and the goals will follow?

James Kiss

Numerade Educator

Problem 19

Tracking hurricanes 2012 In Chapter $6,$ we looked at data from the National Oceanic and Atmospheric Administration about their success in predicting hurricane tracks. On the following page is a scatterplot of the error (in nautical miles) for predicting hurricane locations 24 hours in the future vs. the year in which the prediction (and the hurricane) occurred. In Chapter $6,$ we could describe this relationship only in general terms. Now we can learn more. Here is the regression analysis:
Response variable is 24 Error R-squared $=72.5 \%$ R-squared (adjusted) $=71.8 \%$ $s=16.13$ with $43-2=41$ degrees of freedom $\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 133.024 & 4.835 & 27.5 & \leq 0.0001 \\ \text { rs Since 1970 } & -2.06016 & 0.1982 & -10.4 & \leq 0.0001\end{array}$ Years
a) Explain in words and numbers what the regression says.
b) State the hypothesis about the slope (both numerically and in words) that describes how hurricane prediction quality has changed.
c) Assuming that the assumptions for inference are satisfied, perform the hypothesis test and state your conclusion. Be sure to state it in terms of prediction errors and years.
d) Explain what the $R$ -squared means in terms of this regression.

James Kiss

Numerade Educator

Problem 20

Drug use 2013 The 2013 World Drug Report investigated the prevalence of drug use as a percentage of the population aged 15 to 64. Data from 32 European countries are shown in the following scatterplot and regression analysis. (Source: World Drug Report, 2013. www.unodc .org/unodc/en/data-and-analysis/WDR-2013.html) Response variable is Cocaine R-squared $=25.8 \%$ $s=0.7076$
$32-2=30$ degrees of freedom with
$\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 0.296899 & 0.2627 & 1.13 & 0.2673 \\ \text { Canabis } & 0.123638 & 0.0382 & 3.23 & 0.0030\end{array}$ a) Explain in context what the regression says.
b) State the hypothesis about the slope (both numerically and in words) that describes how use of marijuana is associated with other drugs.
c) Assuming that the assumptions for inference are satisfied, perform the hypothesis test and state your conclusion in context.
d) Explain what $R$ -squared means in context.
e) Do these results indicate that marijuana use leads to the use of harder drugs? Explain.

James Kiss

Numerade Educator

Problem 21

Movie budgets ${ }^{13}$ How does the cost of a movie depend on its length? Data on the cost (millions of dollars) and the running time (minutes) for major release films of 2005 are summarized in these plots and computer output: Dependent variable is Budget(\$million) R-squared $=27.3 \%$ $\mathrm{s}=32.95$ with 1
$120-2=118$ degrees of freedom
$\begin{array}{lccrr}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -63.9981 & 17.12 & -3.74 & 0.0003 \\ \text { Run Time } & 1.02648 & 0.1540 & 6.66 & \leq 0.0001\end{array}$ a) Explain in context what the regression says.
b) The intercept is negative. Discuss its value.
c) The output reports $s=32.95 .$ Explain what that means in this context.
d) What's the value of the standard error of the slope of the regression line?
e) Explain what that means in this context.

James Kiss

Numerade Educator

Problem 22

Saratoga house prices How does the price of a house depend on its size? Data from Saratoga, New York, on 1064 randomly selected houses that had been sold include data on price $(\$ 1000 \mathrm{~s})$ and size $\left(\$ 1000 \mathrm{~s} \mathrm{ft}^{2}\right),$ producing the following graphs and computer output: a) Explain in context what the regression says.
b) The intercept is negative. Discuss its value, taking note of its $\mathrm{P}$ -value.
c) The output reports $s=53.79 .$ Explain what that means in this context.
d) What's the value of the standard error of the slope of the regression line?
e) Explain what that means in this context.

James Kiss

Numerade Educator

Problem 23

Movie budgets, the sequel Exercise 21 shows computer output examining the association between the length of a movie and its cost.
a) Check the assumptions and conditions for inference.
b) Find a $95 \%$ confidence interval for the slope and interpret it in context.

James Kiss

Numerade Educator

Problem 24

Second home Exercise 22 shows computer output examining the association between the sizes of houses and their sale prices.
a) Check the assumptions and conditions for inference.
b) Find a $95 \%$ confidence interval for the slope and interpret it in context.

James Kiss

Numerade Educator

Problem 25

Hot dogs Healthy eating probably doesn't include hot dogs, but if you are going to have one, you'd probably hope it's low in both calories and sodium. Recently, Consumer Reports listed the number of calories and sodium content (in milligrams) for 13 brands of all-beef hot dogs it tested. Examine the association, assuming that the data satisfy the conditions for inference.
Dependent variable is Sodium R-squared $=60.5 \%$ $s=59.66$ with $13-2=11$ degrees of freedom
$\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Constant } & 90.9783 & 77.69 & 1.17 & 0.2663 \\ \text { Calories } & 2.29959 & 0.5607 & 4.10 & 0.0018\end{array}$
a) State the appropriate hypotheses about the slope.
b) Test your hypotheses and state your conclusion in the proper context.

James Kiss

Numerade Educator

Problem 26

Cholesterol Does a person's cholesterol level tend to change with age? Data collected from 1406 adults aged 45 to 62 produced the regression analysis shown. Assuming that the data satisfy the conditions for inference, examine the association between age and cholesterol level.
Dependent variable is Chol $s=46.16$
$\begin{array}{llccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 194.232 & 13.55 & 14.3 & \leq 0.0001 \\ \text { Age } & 0.771639 & 0.2574 & 3.00 & 0.0056\end{array}$
a) State the appropriate hypothesis for the slope.
b) Test your hypothesis and state your conclusion in the proper context.

James Kiss

Numerade Educator

Problem 27

Second frank Look again at Exercise 25's regression output for the calorie and sodium content of hot dogs.
a) The output reports $s=59.66$. Explain what that means in this context.
b) What's the value of the standard error of the slope of the regression line?
c) Explain what that means in this context.

James Kiss

Numerade Educator

Problem 28

More cholesterol Look again at Exercise 26's regression output for age and cholesterol level.
a) The output reports $s=46.16 .$ Explain what that means in this context.
b) What's the value of the standard error of the slope of the regression line?
c) Explain what that means in this context.

James Kiss

Numerade Educator

Problem 29

Last dog Based on the regression output seen in Exercise $25,$ create a $95 \%$ confidence interval for the slope of the regression line and interpret your interval in context.

James Kiss

Numerade Educator

Problem 30

Cholesterol, finis Based on the regression output seen in Exercise $26,$ create a $95 \%$ confidence interval for the slope of the regression line and interpret it in context.

James Kiss

Numerade Educator

Problem 31

Marriage age 2011 The scatterplot suggests a decrease in the difference in ages at first marriage for men and women since $1975 .$ We want to examine the regression to see if this decrease is significant.
a) Write appropriate hypotheses.
b) Here is the residuals plot and a histogram of the residuals. Do you think the conditions for inference are satisfied? Explain.
c) Test the hypothesis and state your conclusion about the trend in age at first marriage.

James Kiss

Numerade Educator

Problem 32

Used cars 2010 Vehix.com offered several used Toyota Corollas for sale. Listed below are the ages of the cars and the advertised prices.
a) Make a scatterplot for these data.
b) Do you think a linear model is appropriate? Explain.
c) Find the equation of the regression line.
d) Check the residuals to see if the conditions for inference are met.

James Kiss

Numerade Educator

Problem 33

Marriage age 2011, again Based on the analysis of marriage ages since 1975 given in Exercise $31,$ find a $95 \%$ confidence interval for the rate at which the age gap is closing. Explain what your confidence interval means.

James Kiss

Numerade Educator

Problem 34

Used cars 2010 , again Based on the analysis of used car prices you did for Exercise $32,$ create a $95 \%$ confidence interval for the slope of the regression line and explain what your interval means in context.

James Kiss

Numerade Educator

Problem 35

Fuel economy A consumer organization has reported test data for 50 car models. We will examine the association between the weight of the car (in thousands of pounds) and the fuel efficiency (in miles per gallon). Here are the scatterplot, summary statistics, and regression analysis:
a) Is there strong evidence of an association between the weight of a car and its gas mileage? Write an appropriate hypothesis.
b) Are the assumptions for regression satisfied?
c) Test your hypothesis and state your conclusion.

James Kiss

Numerade Educator

Problem 36

SAT scores How strong was the association between student scores on the Math and Verbal sections of the old SAT? Scores on each ranged from 200 to 800 and were widely used by college admissions offices. Here are summaries and plots of the scores for a graduating class at Ithaca High School:
a) Is there evidence of an association between Math and Verbal scores? Write an appropriate hypothesis.
b) Discuss the assumptions for inference.
c) Test your hypothesis and state an appropriate conclusion.

James Kiss

Numerade Educator

Problem 37

Fuel economy, part II Consider again the data in Exercise 35 about the gas mileage and weights of cars.
a) Create a $95 \%$ confidence interval for the slope of the regression line.
b) Explain in this context what your confidence interval means.

James Kiss

Numerade Educator

Problem 38

SATs, part II Consider the high school SAT scores data from Exercise 36 .
a) Find a $90 \%$ confidence interval for the slope of the true line describing the association between Math and Verbal scores.
b) Explain in this context what your confidence interval means.

James Kiss

Numerade Educator

Problem 39

Fuel economy, part III Consider again the data in Exercise 35 about the gas mileage and weights of cars.
a) Create a $95 \%$ confidence interval for the average fuel efficiency among cars weighing 2500 pounds, and explain what your interval means.
b) Create a $95 \%$ prediction interval for the gas mileage you might get driving your new 3450 -pound SUV, and explain what that interval means.

James Kiss

Numerade Educator

Problem 40

SATs, again Consider the high school SAT scores data from Exercise 36 once more.
a) Find a $90 \%$ confidence interval for the mean SAT-Math score for all students with an SAT-Verbal score of 500 .
b) Find a $90 \%$ prediction interval for the Math score of the senior class president if you know she scored 710 on the Verbal section.

James Kiss

Numerade Educator

Problem 41

Cereals A healthy cereal should be low in both calories and sodium. Data for 77 cereals were examined and judged acceptable for inference. The 77 cereals had between 50 and 160 calories per serving and between 0 and $320 \mathrm{mg}$ of sodium per serving. Here's the regression analysis:
Dependent variable is Sodium R-squared $=9.0 \%$ $s=80.49$ with $77-2=75$ degrees of freedom
$\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 21.4143 & 51.47 & 0.416 & 0.6786 \\ \text { Calories } & 1.29357 & 0.4738 & 2.73 & 0.0079\end{array}$
a) Is there an association between the number of calories and the sodium content of cereals? Explain.
b) Do you think this association is strong enough to be useful? Explain.

James Kiss

Numerade Educator

Problem 42

Brain size Does your IQ depend on the size of your brain? A group of female college students took a test that measured their verbal IQs and also underwent an MRI scan to measure the size of their brains (in $1000 \mathrm{~s}$ of pixels). The scatterplot and regression analysis are shown, and the assumptions for inference were satisfied.
a) Test an appropriate hypothesis about the association between brain size and $\mathrm{IQ}$.
b) Stale your conclusion about the strength of this association.

James Kiss

Numerade Educator

Problem 43

Cereals, part 2 Further analysis of the data for the breakfast cereals in Exercise 41 looked for an association between Fiber content and Calories by attempting to construct a linear model. Here are several graphs. Which of the assumptions for inference are violated? Explain.

James Kiss

Numerade Educator

Problem 44

Winter The output shows an attempt to model the association between average January Temperature (in degrees Fahrenheit) and Latitude (in degrees north of the equator) for 59 U.S. cities. Which of the assumptions for inference do you think are violated? Explain.

James Kiss

Numerade Educator

Problem 45

Streams Biologists studying the effects of acid rain on wildlife collected data from 172 streams in the Adirondack Mountains. They recorded the $p H$ (acidity) of the water and the $B C I$, a measure of biological diversity. Here's a scatterplot of $B C I$ against $p H$ for the 163 streams for which we have these data:
And here is part of the regression analysis:
Dependent variable is $\mathrm{BCl}$ R-squared $=27.1 \%$ $s=140.4$ with $163-2=161$ degrees of freedom
$\begin{array}{lcc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } \\ \text { Intercept } & 2733.37 & 187.9 \\ \mathrm{pH} & -197.694 & 25.57\end{array}$
a) State the null and alternative hypotheses under investigation.
b) Assuming that the assumptions for regression inference are reasonable, find the $t$ - and P-values.
c) State your conclusion.

James Kiss

Numerade Educator

Problem 46

Climate change 2013 Data collected from around the globe show that the earth is getting warmer. The most common theory relates climate change to an increase in atmospheric levels of carbon dioxide $\left(\mathrm{CO}_{2}\right)$, a greenhouse gas. The mean annual $\mathrm{CO}_{2}$ concentration in the atmosphere (parts per million) is measured at the top of Mauna Loa in Hawaii (away from any local contaminants) and available at ftp://aftp.cmdl.noaa.gov /products/trends/co2/co2_annmean_mlo.txt
The mean surface air temperature is recorded as the change in ${ }^{\circ} \mathrm{C}$ relative to a base period of 1951 to $1980 .$ It is available at data.giss.nasa.gov/gistemp/graphs_v3/
Here are a scatterplot and regression for the years from 1970 to 2013: Response variable is Global Avg Temp R-squared $=73.6 \% \quad$ R-squared (adjusted) $=72.9 \%$ $s=0.1316$ with $44-2=42$ degrees of freedom
$\begin{array}{lcccl}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 10.6442 & 0.3418 & 31.1 & \leq 0.0001 \\ \mathrm{C} 0_{2} & 0.010325 & 0.0010 & 10.8 & \leq 0.0001\end{array}$
a) Write the equation of the regression line.
b) Is there evidence of an association between $\mathrm{CO}_{2}$ level and global temperature?
c) Do you think predictions made by this regression will be very accurate? Explain.
d) Does this regression prove that increasing $\mathrm{CO}_{2}$ levels are causing global warming? Discuss.

James Kiss

Numerade Educator

Problem 47

Ozone and population The Environmental Protection Agency is examining the relationship between the ozone level (in parts per million) and the population (in millions) of U.S. cities. Part of the regression analysis is shown. Dependent variable is 0zone R-squared $=84.4 \%$ h $16-2=14$ df $s=5.454$ with
Variable $-$ Coe efficie SE(Coeff Intercept $\begin{array}{cc}18.892 & 2.395\end{array}$ Pop 6.650 1.910
a) We suspect that the greater the population of a city, the higher its ozone level. Is the relationship statistically significant? Assuming the conditions for inference are satisfied, test an appropriate hypothesis and state your conclusion in context.
b) Do you think that the population of a city is a useful predictor of ozone level? Use the values of both $R^{2}$ and $s$ in your explanation.

James Kiss

Numerade Educator

Problem 48

Sales and profits A business analyst was interested in the relationship between a company's sales and its profits. She collected data (in millions of dollars) from a random sample of Fortune 500 companies and created the regression analysis and summary statistics shown. The assumptions for regression inference appeared to be satisfied. \begin{tabular}{lll|lrc}
& \multicolumn{1}{l} { Profits } & \multicolumn{1}{c|} { Sales } & \multicolumn{2}{l} { Dependent variable is Profits } \\
Count & 79 & 79 & R-squared $=66.2 \%$ & $s=466.2$ \\
Mean & 209.839 & 4178.29 & Variable & Coefficient & SE(Coeff) \\
Variance & 635,172 & 49,163,000 & Intercept & -176.644 & 61.16 \\
Std Dev & 796.977 & 7011.63 & Sales & 0.092498 & 0.0075
\end{tabular}
a) Is there a statistically significant association between sales and profits? Test an appropriate hypothesis and state your conclusion in context.
b) Do you think that a company's sales serve as a useful predictor of its profits? Use the values of both $R^{2}$ and $s$ in your explanation.

James Kiss

Numerade Educator

Problem 49

Ozone, again Consider again the relationship between the population and ozone level of U.S. cities that you analyzed in Exercise 47
a) Give a $90 \%$ confidence interval for the approximate increase in ozone level associated with each additional
million city inhabitants.
b) For the cities studied, the mean population was
1. 7 million people. The population of Boston is approximately 0.6 million people. Predict the mean ozone level for cities of that size with an interval in which you have $90 \%$ confidence.

James Kiss

Numerade Educator

Problem 50

More sales and profits Consider again the relationship between the sales and profits of Fortune 500 companies that you analyzed in Exercise 48 .
a) Find a $95 \%$ confidence interval for the slope of the regression line. Interpret your interval in context.
b) Last year, the drug manufacturer Eli Lilly, Inc., reported gross sales of $\$ 23$ billion (that's $\$ 23,000$ million). Create a $95 \%$ prediction interval for the company's profits, and interpret your interval in context.

James Kiss

Numerade Educator

Problem 51

Tablet computers 2014 cnet.com tests tablet computers and continuously updates its list. As of January 2014 , the list included the battery life (in hours) and luminous intensity (i.e., screen brightness, in $\mathrm{cd} / \mathrm{m}^{2}$ ). We want to know if brighter screens are related to the maximum Screen Brightness. (http://reviews.cnet.com $/ 8301-19736_{-} 7-20080768-251 /$ cnet-updates-tablet-testresults/?tag=contentBody;contentHighlights) Response variable is $\quad$ Battery life (hrs) R-squared $=11.3 \%$ $s=2.128$ with $34-2=32$ degrees of freedom $\begin{array}{lllll}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 5.38719 & 1.727 & 3.12 & 0.0038 \\ \text { Max Brightness } & 0.00904 & 0.0045 & 2.02 & 0.0522\end{array}$
a) How many tablet computers were tested?
b) Are the conditions for inference satisfied? Explain.
c) Is there evidence of an association between maximum brightness of the screen and battery life? Test an appropriate hypothesis and state your conclusion.
d) Is the association strong? Explain.
e) What is the equation of the regression line?
f) Create a $90 \%$ confidence interval for the slope of the true line.
g) Interpret your interval in this context.

James Kiss

Numerade Educator

Problem 52

Crawling Researchers at the University of Denver Infant Study Center wondered whether temperature might influence the age at which babies learn to crawl. Perhaps the extra clothing that babies wear in cold weather would restrict movement and delay the age at which they started crawling. Data were collected on 208 boys and 206 girls. Parents reported the month of the baby's birth and the age (in weeks) at which their child first crawled. The table gives the average Temperature $\left({ }^{\circ} \mathrm{F}\right)$ when the babies were 6 months old and average Crawling Age (in weeks) for each month of the year. Make the plots and compute the analyses necessary to answer the following questions. a) Would this association appear to be weaker, stronger, or the same if data had been plotted for individual babies instead of using monthly averages? Explain.
b) Is there evidence of an association between Temperature and Crawling Age? Test an appropriate hypothesis and state your conclusion. Don't forget to check the assumptions.
c) Create and interpret a $95 \%$ confidence interval for the slope of the true relationship.

James Kiss

Numerade Educator

Problem 53

Body fat Do the data shown in the table below indicate an association between Waist size and \%Body Fat?
a) Test an appropriate hypothesis and state your conclusion.
b) Give a $95 \%$ confidence interval for the mean $\%$ Body Fat found in people with a 40 -inch Waist.

Rashmi Sinha

Numerade Educator

Problem 54

Body fat, again Use the data from Exercise 53 to examine the association between Weight and \%Body Fat.
a) Find a $90 \%$ confidence interval for the slope of the regression line of \%Body Fat on Weight.
b) Interpret your interval in context.
c) Give a $95 \%$ prediction interval for the \%Body Fat of an individual who weighs 165 pounds.

James Kiss

Numerade Educator

Problem 55

Grades The data set below shows midterm scores from an Introductory Statistics course.
a) Fit a model predicting the second midterm score from the first.
b) Comment on the model you found, including a discussion of the assumptions and conditions for regression. Is the coefficient for the slope statistically significant?
c) A student comments that because the P-value for the slope is very small, Midterm 2 is very well predicted from Midterm 1. So, he reasons, next term the professor can give just one midterm. What do you think?

James Kiss

Numerade Educator

Problem 56

Grades? The professor teaching the Introductory Statistics class discussed in Exercise 55 wonders whether performance on homework can accurately predict midterm scores.
a) To investigate it, she fits a regression of the sum of the two midterms scores on homework scores. Fit the regression model.
b) Comment on the model including a discussion of the assumptions and conditions for regression. Is the coefficient for the slope "statistically significant"?
c) Do you think she can accurately judge a student's performance without giving the midterms? Explain.

James Kiss

Numerade Educator

Problem 57

Strike two Remember the Little League instructional video discussed in Chapter $23 ?$ Ads claimed it would improve the performances of Little League pitchers. To test this claim, 20 Little Leaguers threw 50 pitches each, and we recorded the number of strikes. After the players participated in the training program, we repeated the test. The table shows the number of strikes each player threw before and after the training. A test of paired differences failed to show that this training improves ability to throw strikes. Is there any evidence that the effectiveness of the video (After - Before) depends on the player's initial ability to throw strikes (Before)? Test an appropriate hypothesis and state your conclusion. Propose an explanation for what you find.

James Kiss

Numerade Educator

Problem 58

All the efficiency money can buy 2011 A sample of 84 model- 2011 cars from an online information service was examined to see how fuel efficiency (as highway mpg) relates to the cost (Manufacturer's Suggested Retail Price in dollars) of cars. Here are displays and computer output:
a) State what you want to know, identify the variables, and give the appropriate hypotheses.
b) Check the assumptions and conditions.
c) If the conditions are met, complete the analysis.

James Kiss

Numerade Educator

Problem 59

Education and mortality The following software output is based on the mortality rate (deaths per 100,000 people) and the education level (average number of years in school) for 58 U.S. cities.
\begin{tabular}{lccc}
Variable & Count & Mean & StdDev \\
Mortality & 58 & 942.501 & 61.8490 \\
Education & 58 & 11.0328 & 0.793480 \\
\hline
\end{tabular} Dependent variable is Mortality $R$ -squared $=41.0 \%$ $s=47.92$ with $58-2=56$ degrees of freedom Variable
Intercept
Education $\begin{array}{cc}\text { Coefficient } & \text { SE(Coeff) } \\ 1493.26 & 88.48 \\ -49.9202 & 8.000\end{array}$
a) Comment on the assumptions for inference.
b) Is there evidence of a strong association between the level of Education in a city and the Mortality rate? Test an appropriate hypothesis and state your conclusion.
c) Can we conclude that getting more education is likely (on average) to prolong your life? Why or why not?
d) Find a $95 \%$ confidence interval for the slope of the true relationship.
e) Explain what your interval means.
f) Find a $95 \%$ confidence interval for the average Mortality rate in cities where the adult population completed an average of 12 years of school.

James Kiss

Numerade Educator

Problem 60

Property assessments The following software outputs provide information about the Size (in square feet) of 18 homes in Ithaca, New York, and the city's assessed Value of those homes. \begin{tabular}{lcccc}
Variable & Count & Mean & StdDev & Range \\
Size & 18 & 2003.39 & 264.727 & 890 \\
Value & 18 & 60946.7 & 5527.62 & 19710 \\
\hline
\end{tabular}
Dependent variable is Value R-squared $=32.5 \%$ $s=4682$ with $18-2=16$ degrees of freedom $\begin{array}{lcc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } \\ \text { Intercept } & 37108.8 & 8664 \\ \text { Size } & 11.8987 & 4.290\end{array}$
a) Explain why inference for linear regression is appropriate with these data.
b) Is there a significant association between the Size of a home and its assessed Value? Test an appropriate hypothesis and state your conclusion.
c) What percentage of the variability in assessed Value is explained by this regression?
d) Give a $90 \%$ confidence interval for the slope of the true regression line, and explain its meaning in the proper context.
e) From this analysis, can we conclude that adding a room to your house will increase its assessed Value? Why or why not?
f) The owner of a home measuring 2100 square feet files an appeal, claiming that the $\$ 70,200$ assessed Value is too high. Do you agree? Explain your reasoning.

Meri Siavoshi

Numerade Educator

Problem 61

Right-to-work laws Are so-called "right-to-work" laws related to the percent of public sector employees in unions and the percent of private sector employees in unions? This data set looks at these percentages for the states in the United States in $1982 .$ The dependent variable is whether the state had a right-to-work law or not. The computer output for the logistic regression is given here. (Source: N. M. Meltz, "Interstate and Interprovincial Differences in Union Density," Industrial Relations, 28: 2 [Spring 1989$], 142-158$ by way of DASL.)
Logistic Regression Table
\begin{tabular}{lllrl}
Predictor & \multicolumn{1}{l} { Coeff } & SE(Coeff) & Chisq & $\mathrm{P}$ \\
Intercept & 6.1995 & 1.78724 & 12.04 & 0.001 \\
publ & -0.1062 & 0.0475 & 5.02 & 0.025 \\
pvt & -0.2230 & 0.0811 & 7.56 & 0.006
\end{tabular}
a) Write out the estimated regression equation.
b) The following are scatterplots of the response variable against each of the explanatory variables. Does logistic regression seem appropriate here? Explain.

James Kiss

Numerade Educator

Problem 62

Cost of higher education Are there fundamental differences between liberal arts colleges and universities? In this case, we have information on the top 25 liberal arts colleges and the top 25 universities in the Unites States. We will consider the type of school as our response variable and will use the percent of students who were in the top $10 \%$ of their high school class and the amount of money spent per student by the college or university as our explanatory variables. The output from this logistic regression is given on the following page. \begin{tabular}{lrlrc}
Predictor & \multicolumn{1}{c} { Coeff } & SE(Coeff) & Chisq & $\mathrm{P}$ \\
Intercept & -13.1461 & 3.98629 & 10.89 & 0.001 \\
Top $10 \%$ & 0.0845 & 0.0396345 & 4.54 & 0.033 \\
\$/Student & 0.0003 & 0.0000860 & 9.12 & 0.003
\end{tabular}
a) Write out the estimated regression equation.
b) Is percent of students in the top $10 \%$ of their high school class statistically significant in predicting whether or not the school is a university? Explain.
c) Is the amount of money spent per student statistically significant in predicting whether or not the school is a university? Explain.

James Kiss

Numerade Educator