Chapter 13.3, Problem 26PB

Any predictive power? Refer to the previous three exercises.

1. a. State and interpret the null hypothesis tested with the F statistic in the ANOVA table given in Exercise 13.23.
2. b. From the F table (Table D), which F statistic value would have a P-value of 0.05 for these data?
3. c. Report the observed F test statistic and its P-value. Interpret the P-value, and make a decision for a 0.05 significance level. Explain in nontechnical terms what the result of the test means.

13.23 Does leg press help predict body strength? Chapter 12 analyzed strength data for 57 female high school athletes. Upper body strength was summarized by the maximum number of pounds the athlete could bench press (denoted maxBP). This was predicted well by the number of times she could do a 60-pound bench press (denoted BP60). Can we predict maxBP even better if we also know how many times an athlete can perform a 200-pound leg press? The table shows results after adding this second predictor (denoted LP200) to the model.

1. a. Does LP200 have a significant effect on maxBP if BP60 is also in the model? Show all steps of a significance test to answer this.
2. b. Show that the 95% confidence interval for the slope for LP200 equals 0.21 ± 0.30, roughly (−0.1, 0.5). Based on this interval, does LP200 seem to have a strong impact, or a weak impact, on predicting maxBP if BP60 is also in the model?
3. c. Given that LP200 is in the model, provide evidence from a significance test that shows why it does help to add BP60 to the model.

13.24 Leg press uncorrelated with strength? The P-value of 0.17 in part a of the previous exercise suggests that LP200 plausibly had no effect on maxBP once BP60 is in the model. Yet when LP200 is the sole predictor of BP, the correlation is 0.58 and the significance test for its effect has a P-value of 0.000, suggesting very strong evidence of an effect. Explain why this is not a contradiction.

13.25 Interpret strength variability Refer to the previous two exercises. The sample standard deviation of maxBP was 13.3. The residual standard deviation of maxBP when BP60 and LP200 are predictors in a multiple regression model is 7.9.

1. a. Explain the difference between the interpretations of these two standard deviations.
2. b. If the conditional distributions of maxBP are approximately bell shaped, explain why most maximum bench press values fall within about 16 pounds of the regression equation when the predictors BP60 and LP200 are near their sample mean values.
3. c. At BP60 = 11 and LP200 = 22, which are close to the sample mean values, software reports $\hat{y}$ = 80 and a 95% prediction interval of 80 ± 16, or (64, 96). Is this interval an inference about where the population maxBP values fall or where the population mean of the maxBP values fall (for subjects having BP60 = 11 and LP200 = 22)? Explain.
4. d. Refer to part c. Would it be unusual for a female athlete with these predictor values to be able to bench press more than 100 pounds? Why?