SECTION 10.2 EXERCISES

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For Exercise 10.23, see page 587; and for Exercise 10.24, see page 590.

Question 10.25

10.25 What’s wrong? For each of the following, explain what is wrong and why.

  1. (a) In simple linear regression, the null hypothesis of the ANOVA F test is H0: β0 = 0.

  2. (b) In an ANOVA table, the mean squares add. In other words, MST = MSM + MSE.

  3. (c) The smaller the P-value for the ANOVA F test, the greater the explanatory power of the model.

  4. (d) The total degrees of freedom in an ANOVA table are equal to the number of observations n.

Question 10.26

10.26 What’s wrong? For each of the following, explain what is wrong and why.

  1. (a) In simple linear regression, the standard error for a future observation is s, the measure of spread about the regression line.

  2. (b) In an ANOVA table, SSE is the sum of the deviations.

  3. (c) There is a close connection between the correlation r and the intercept of the regression line.

  4. (d) The squared correlation r2 is equal to MSM/MST.

Question 10.27

10.27 Research and development obligations. The National Science Foundation collects data on the research and development obligations for science and engineering to universities and colleges in the United States.10 Here are the data for the years 2006, 2008, 2010, and 2012:

Year 2006 2008 2010 2012
Spending (billions of dollars) 31.0 35.2 39.4 41.7

Do the following by hand or with a calculator and verify your results with a software package of Excel.

  1. (a) Make a scatterplot that shows the increase in research and development obligations over time. Does the pattern suggest that the obligations are increasing linearly over time?

  2. (b) Find the equation of the least-squares regression line for predicting obligations from year. Add this line to your scatterplot.

  3. (c) For each of the four years, find the residual. Use these residuals to calculate the estimated model standard error s.

  4. (d) Write the regression model for this setting. What are your estimates of the unknown parameters in this model?

  5. (e) Compute a 95% confidence interval for the slope and summarize what this interval tells you about the increase in obligations over time.

Question 10.28

10.28 Food neophobia. Food neophobia is a personality trait associated with avoiding unfamiliar foods. In one study of 564 children who were two to six years of age, food neophobia and the frequency of consumption of different types of food were measured.11 Here is a summary of the correlations:

Type of food Correlation
Vegetables −0.27
Fruit −0.16
Meat −0.15
Eggs −0.08
Sweet/fatty snacks 0.04
Starchy staples −0.02

Perform the significance test for each correlation and write a summary about food neophobia and the consumption of different types of food.

Question 10.29

10.29 Correlation between the prevalences of adult binge drinking and underage drinking. A group of researchers compiled data on the prevalence of adult binge drinking and the prevalence of underage drinking in 42 states.12 A correlation of 0.32 was reported.

  1. (a) Test the null hypothesis that the population correlation ρ = 0 against the alternative ρ > 0. Are the results significant at the 5% level?

  2. (b) Explain this correlation in terms of the direction of the association and the percent of variability in the prevalence of underage drinking that is explained by the prevalence of adult binge drinking.

  3. (c) The researchers collected information from 42 of 50 states, so almost all the data available was used in the analysis. Provide an argument for the use of statistical inference in this setting.

Question 10.30

10.30 Grade inflation. The average undergraduate GPA for American colleges and universities was estimated based on a sample of institutions that published this information.13 Here are the data for public schools in that report:

Year 1992 1996 2002 2007
GPA 2.85 2.90 2.97 3.01

Do the following by hand or with a calculator and verify your results with a software package.

  1. (a) Make a scatterplot that shows the increase in GPA over time. Does a linear increase appear reasonable?

  2. (b) Find the equation of the least-squares regression line for predicting GPA from year. Add this line to your scatterplot.

  3. (c) Compute a 95% confidence interval for the slope and summarize what this interval tells you about the increase in GPA over time.

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Question 10.31

10.31 Completing an ANOVA table. How are returns on common stocks in overseas markets related to returns in U.S. markets? Consider measuring U.S. returns by the annual rate of return on the Standard & Poor’s 500 stock index and overseas returns by the annual rate of return on the Morgan Stanley Europe, Australasia, Far East (EAFE) index.14 Both are recorded in percents. We will regress the EAFE returns on the S&P 500 returns for the years 1989 to 2014. Here is part of the Minitab output for this regression:

The regression equation is

EAFE = −3.19 + 0.813 S&P

Analysis of Variance

Source    DF    SS  MS  F

Regression  1  5552.9

Residual Error

Total       10077.9

Using the ANOVA table format on page 586 as a guide, complete the analysis of variance table.

Question 10.32

10.32 Interpreting statistical software output. Refer to the previous exercise. What are the values of the estimated model standard error s and the squared correlation r2?

Question 10.33

image 10.33 Confidence intervals for the slope and intercept. Refer to the previous two exercises. The mean and standard deviation of the S&P 500 returns for these years is 12.04% and 18.33%, respecitvely. From this and your work in the previous exercise:

  1. (a) Find the standard error for the least-squares slope b1.

  2. (b) Give a 95% confidence interval for the slope β1 of the population regression line.

  3. (c) Explain why the intercept β0 is meaningful in this example.

  4. (d) Find the standard error for the least-squares intercept b0 and use it to construct a 95% confidence interval.