A multimedia statistics learning system includes a test of skill in using the computer's mouse. The software displays a circle at a random location on the computer screen. The subject clicks in the circle with the mouse as quickly as possible. A new circle appears as soon as the subject clicks in the old one. The table reaction times (in milliseconds) in a computer game gives data for one subject's trials, 20 with each hand. Distance is the distance from the cursor location to the center of the new circle, in units whose actual size depends on the size of the screen. Time is the time required to click in the new circle, in milliseconds. We suspect that time depends on distance. We also suspect that performance will not be the same with the right and left hands. Analyze the data with a view to predicting performance separately for the two hands.

$$\begin{array}{ccc|ccc} \text { TIME } & \text { DISTANCE } & \text { HAND } & \text { TIME } & \text { DISTANCE } & \text { HAND } \\ \hline 115 & 190.70 & \text { right } & 240 & 190.70 & \text { left } \\ \hline 96 & 138.52 & \text { right } & 190 & 138.52 & \text { left } \\ \hline 110 & 165.08 & \text { right } & 170 & 165.08 & \text { left } \\ \hline 100 & 126.19 & \text { right } & 125 & 126.19 & \text { left } \\ \hline 111 & 163.19 & \text { right } & 315 & 163.19 & \text { left } \\ \hline 101 & 305.66 & \text { right } & 240 & 305.66 & \text { left } \\ \hline 111 & 176.15 & \text { right } & 141 & 176.15 & \text { left } \\ \hline 106 & 162.78 & \text { right } & 210 & 162.78 & \text { left } \\ \hline 96 & 147.87 & \text { right } & 200 & 147.87 & \text { left } \\ \hline 96 & 271.46 & \text { right } & 401 & 271.46 & \text { left } \\ \hline 95 & 40.25 & \text { right } & 320 & 40.25 & \text { left } \\ \hline 96 & 24.76 & \text { right } & 113 & 24.76 & \text { left } \\ \hline 96 & 104.80 & \text { right } & 176 & 104.80 & \text { left } \\ \hline 106 & 136.80 & \text { right } & 211 & 136.80 & \text { left } \\ \hline 100 & 308.60 & \text { right } & 238 & 308.60 & \text { left } \\ \hline 113 & 279.80 & \text { right } & 316 & 279.80 & \text { left } \\ \hline 123 & 125.51 & \text { right } & 176 & 125.51 & \text { left } \\ \hline 111 & 329.80 & \text { right } & 173 & 329.80 & \text { left } \\ \hline 95 & 51.66 & \text { right } & 210 & 51.66 & \text { left } \\ \hline 108 & 201.95 & \text { right } & 170 & 201.95 & \text { left } \\ \hline \end{array}$$

In the earlier exercise, you used the Correlation and Regression applet to create three scatterplots having a correlation of about r = 0. 7 between the horizontal variable x and the vertical variable y. Create three similar scatterplots again, and click the "Show least-squares line" box to display the regression lines. Correlation r = 0. 7 is considered reasonably strong in many areas of work. Because there is a reasonably strong correlation, we might use a regression line to predict y from x. In which of your three scatterplots does it make sense to use a straight line for prediction?

We expect that students who do well on the midterm exam in a course will usually also do well on the final exam. Gary Smith of Pomona College looked at the exam scores of all 346 students who took his statistics class over a I0-year period. The least-squares line for predicting final exam score from midterm-exam score was $\hat{y}$ = 46.6 + 0.41x. (Both exams have a 100-point scale.)

Octavio scores 10 points above the class mean on the midterm. How many points above the class mean do you predict that he will score on the final? (Hint: Use the fact that the least-squares line passes through the point ($\bar{x}, \bar{y}$) and the fact that Octavio's midterm score is $\bar{x}$ + 10.) This is another example of regression to the mean: students who do well on the midterm will, in general, do less well, but still above average, on the final.

In the previous figure it displays the relationship between golfers' scores on the first and second rounds of the 2010 Masters Tournament. The correlation is $r=0.347$. Exercise 4.30 gives data on solar radiation (SRD) and concentration of dimethyl sulfide (DMS) over a region of the Mediterranean. The correlation is $r=0.969$. Explain in simple language why knowing only these correlations enables you to say that prediction of DMS from SRD by a regression line will be much more accurate than prediction of a golfer's second-round score from his first-round score.

Use the equation of the least-squares regression line of $\hat{y}$ = a + bx to show that the regression line for predicting y from x always passes through the point ($\bar{x}, \bar{y}$). That is, when $\bar{x} = \bar{x}$, the equation gives $\bar{y} = \bar{y}$ .

In the previous figure it displays the relationship between golfers' scores on the first and second rounds of the 2010 Masters Tournament. The least-squares line for predicting second-round scores from first-round scores has equation $\hat{y}=52.74+0.297 x$. Find the predicted second-round scores for a player who shot 80 in the first round and for a player who shot 70. The mean secondround score for all players was 74.48. So a player who does well in the first round is predicted to do less well, but still better than average, in the second round. And a player who does poorly in the first is predicted to do better, but still worse than average, in the second.

The data set "tchsal2" gives the mean Mathematics SAT score and mean salary of teachers in each of the 50 states and the District of Columbia in 2013. It also includes a categorica l variab le, pct. taking, that indicates whether the percentage taking is above 40% (Y) or below 40% (N).

(a) Find the least-squares line for predicting mean Mathematics SAT score from mean teacher salary for only the cases where the percent taking is above 40%. Interpret the slope in the context of the problem.

(b) Find the least-squares line for predicting mean Mathematics SAT score from mean teacher salary for only the cases where the percent taking is below 40%. Interpret the slope in the context of the problem.

(c) If you did the earlier exercise, compare your results with those in part (a) of the earlier exercise. What do you conclude?

The given figure displays the relationship between golfers' scores on the first and second rounds of the 2016 Masters Tournament. The least squares line for predicting second-round scores (y) from first-round scores (x) has equation $\hat{y}$ = 62.91 + 0.164x. Find the predicted second-round scores for a player who shot 80 in the first round and for a player who shot 70. The mean second-round score for all players was 75.02. So, a player who does well in the first round is predicted to do less well, but still better than average, in the second round. In addition, a player who does poorly in the first is predicted to do better, but still worse than average, in the second.

(Comment: This is a regression to the mean. If you select individuals with extreme scores on some measure, they tend to have less extreme scores when measured again. That's because their extreme position is partly merit and partly luck, and the luck will be different next time. Regression to the mean contributes to lots of "effects." The rookie of the year often doesn't do as well the next year; the best player in an orchestral audition may play less well once hired than the runners-up; a student who feels she needs coaching after taking the SAT often doe s better on the next try without coaching.)

The data set "tchsal" gives the mean Mathematics SAT score and mean salary of teachers in each of the 50 states and the District of Columbia in 2013. The correlation between mean Mathematics SAT score and mean teacher salary is -0.2782.

(a) Find the least-squares line for predicting mean Mathematics SAT score from mean teacher salary. Interpret the slope in the context of the problem.

(b) Is it reasonable to conclude that states can increase the mean Mathematics SAT score in their state by reducing teacher salaries? (Hint: Which states have the highest and lowest mean Mathematics SAT scores? Which states have the highest and lowest cost of living?)

Many colleges offer online versions of courses that are also taught in the classroom. It often happens that the students who enroll in the online version do better than the classroom students on the course exams. This does not show that online instruction is more effective than classroom teaching, because the people who sign up for online courses are often quite different from the classroom students. Suggest some differences between online and classroom students that might explain why online students do better.

Researchers analyzed data from more than 5000 adults and found that the more diet sodas a person drank, the greater their weight gain. Does this mean that drinking diet soda causes weight gain? Give a more plausible explanation for this association.

Homicide and suicide are both intentional means of ending a life. However, the reason for committing homicide is different from that for suicide, and we might expect homicide and suicide rates to be uncorrelated. On the other hand, both can involve some degree of violence, so perhaps we might expect some level of correlation in the rates. Here are data from 2008-2011 for 26 counties in Ohio. Rates are per 100,000 people.

$$\begin{array}{lrr|lrr} \hline \text { County } & \begin{array}{l} \text { Homicide } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Suicide } \\ \text { Rate } \end{array} & \text { County } & \begin{array}{l} \text { Homicide } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Suicide } \\ \text { Rate } \end{array} \\ \hline \text { Allen } & 3.8 & 9.9 & \text { Lorain } & 3.6 & 15.6 \\ \hline \text { Ashtabula } & 2.8 & 15.5 & \text { Lucas } & 7.6 & 12.8 \\ \hline \text { Butler } & 2.7 & 12.8 & \text { Mahoning } & 10.6 & 11.4 \\ \hline \text { Clermont } & 1.0 & 16.0 & \text { Medina } & 1.0 & 10.0 \\ \hline \text { Clark } & 5.4 & 13.6 & \text { Miami } & 1.0 & 9.6 \\ \hline \text { Columbiana } & 3.4 & 15.2 & \text { Montgomery } & 9.5 & 15.5 \\ \hline \text { Cuyahoga } & 10.0 & 9.8 & \text { Portage } & 1.6 & 11.9 \\ \hline \text { Delaware } & 1.0 & 8.2 & \text { Stark } & 4.8 & 13.0 \\ \hline \text { Franklin } & 8.8 & 11.3 & \text { Summit } & 4.9 & 11.5 \\ \hline \text { Greene } & 2.7 & 11.3 & \text { Trumbull } & 6.8 & 16.1 \\ \hline \text { Hamilton } & 9.2 & 10.7 & \text { Warren } & 1.0 & 9.6 \\ \hline \text { Lake } & 1.8 & 12.9 & \text { Wayne } & 1.0 & 10.8 \\ \hline \text { Licking } & 4.0 & 13.2 & \text { Wood } & 1.0 & 8.4 \\ \hline \end{array}$$

(a) Make a scatterplot that shows how the suicide rate can be predicted from the homicide rate. There is a weak linear relationship, with correlation r = 0.17.

(b) Find the least-squares regression line for predicting the suicide rate from the homicide rate. Add this line to your scatterplot.

(c) Explain in words what the slope of the regression line tells us.

(d) Another Ohio county has a homicide rate of 8.0 per 100,000 people. What is the county's predicted suicide rate?

Homicide and suicide are both intentional means of ending a life. However, the reason for committing a homicide is different from that for

$$\begin{array}{lcc|lcc} \begin{array}{l} \text { County } \end{array} & \begin{array}{l} \text { Homicide } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Suicide } \\ \text { Rate } \end{array} & \begin{array}{l} \text { County } \end{array} & \begin{array}{l} \text { Homicide } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Suicide } \\ \text { Rate } \end{array} \\ \text { Allen } & 4.2 & 9.2 & \text { Lorain } & 3.1 & 11.0 \\ \hline \text { Ashtabula } & 1.8 & 15.5 & \text { Lucas } & 7.4 & 13.3 \\ \hline \text { Butler } & 2.6 & 12.7 & \text { Mahoning } & 10.9 & 12.4 \\ \hline \text { Clermont } & 1.0 & 16.0 & \text { Medina } & 0.5 & 10.0 \\ \hline \text { Clark } & 5.6 & 14.5 & \text { Miami } & 2.6 & 9.2 \\ \hline \text { Columbiana } & 3.5 & 16.6 & \text { Montgomery } & 9.5 & 15.2 \\ \hline \text { Cuyahoga } & 9.2 & 9.5 & \text { Portage } & 1.6 & 9.6 \\ \hline \text { Delaware } & 0.8 & 7.6 & \text { Stark } & 4.7 & 13.5 \\ \hline \text { Franklin } & 8.7 & 11.4 & \text { Summit } & 4.9 & 11.5 \\ \hline \text { Greene } & 2.7 & 12.8 & \text { Trumbull } & 5.8 & 16.6 \\ \hline \text { Hamilton } & 8.9 & 10.8 & \text { Warren } & 0.7 & 11.3 \\ \hline \text { Lake } & 1.8 & 11.3 & \text { Wayne } & 1.8 & 8.9 \\ \hline \text { Licking } & 4.5 & 12.9 & \text { Wood } & 1.0 & 7.4 \\ \hline \end{array}$$

suicide and we might expect homicide and suicide rates to be uncorrelated. On the other hand, both can involve some degree of violence, so perhaps we might expect some level of correlation in the rates. The data from 2008-10 for 26 counties in Ohio are shown in the table. Rates are per 100,000 people

(a) Make a scatterplot that shows how suicide rate can be predicted from homicide rate. There is a weak linear relationship, with correlation r = 0.23.

(b) Find the least-squares regression line for predicting suicide rate from homicide rate. Add this line to your scatterplot.

(c) Explain in words what the slope of the regression line tells us.

(d) Another Ohio county has a homicide rate of 8.0 per 100,000 people. What is the county 's predicted suicide rate?

Data on the average income of all men who died in the past year in several U.S. counties showed a positive correlation with the average age of death of men who died in the past year in the counties. Would the correlation be greater, smaller, or about the same if you calculated the correlation between the incomes of individual men who died in the past year and their age at death? Explain your answer.