14.49 Age, hours studied, and regression: In How It Works 13.2, we calculated the correlation coefficient between students’ age and the number of hours they study per week. The mean for age is 21, and the standard deviation is 1.789. The mean for hours studied is 14.2, and the standard deviation is 5.582. The correlation between these two variables is 0.49. Use the z score formula.

a. João is 24 years old. How many hours would we predict he studies per week?

b. Kimberly is 19 years old. How many hours would we predict she studies per week?

c. Seung is 45 years old. Why might it not be a good idea to predict how many hours per week he studies?

d. From a mathematical perspective, why is the word regression used? [Hint: Look at parts

(a) and (b), and discuss the scores on the first variable with respect to their mean versus the predicted scores on the second variable with respect to their mean.]

e. Calculate the regression equation.

f. Use the regression equation to predict the number of hours studied for a 17-year-old student and for a 22-yearold student. g. Using the four pairs of scores that you have [age and predicted hours studied from part (b), and the predicted scores for a score of 0 and 1 from calculating the regression equation], create a graph that includes the regression line. h. Why is it misleading to include young ages such as 0 and 5 on the graph? i. Construct a graph that includes both the scatterplot for these data and the regression line. Draw vertical lines to connect each dot on the scatterplot with the regression line. j. Construct a second graph that includes both the scatterplot and a line for the mean for hours studied, 14.2. The line will be horizontal and will begin at 14.2 on the y-axis. Draw vertical lines to connect each dot on the scatterplot with the line representing the mean. k. Part (i) is a depiction of the error we make if we use the regression equation to predict hours studied. Part (j) is a depiction of the error we make if we use the mean to predict hours studied (i.e., if we predict that everyone has the mean of 14.2 on hours studied per week). Which one appears to have less error? Briefly explain why the error is less in one situation. l. Calculate the proportionate reduction in error the long way. m. Explain what the proportionate reduction in error that you calculated in part (l) tells us. Be specific about what it tells us about making predictions using the regression equation versus making predictions using the mean. n. Demonstrate how the proportionate reduction in error could be calculated using the shortcut. Why does this make sense? That is, why does the correlation coefficient give us a sense of how useful the regression equation will be? o. Compute the standardized regression coefficient. p. How does this coefficient relate to other information you know? q. Draw a conclusion about your analysis based on what you know about hypothesis testing with regression.