Predicting GPA. Continue your study of GPA and high school

Chapter , Problem 16.56

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Predicting GPA. Continue your study of GPA and high school math grades, begun in Exercise 16.52, by performing a regression to predict GPA using high school math grades as the explanatory variable. GPA (a) Plot the residuals against the math grades and make a Normal quantile plot of the residuals. Do these plots suggest that inference based on the usual simple linear regression model may be inaccurate? Give reasons for your answer. (b) Examine the bootstrap distribution of the slope b1 of the least-squares regression line. Based on what you see, what do you recommend regarding the use of bootstrap t or bootstrap percentile intervals? Give reasons for your recommendation. 16-42 CHAPTER 16 Bootstrap Methods and Permutation Tests Significance tests tell us whether an observed effect, such as a difference between two means or a correlation between two variables, could reasonably occur just by chance in selecting a random sample. If not, we have evidence that the effect observed in the sample reflects an effect that is present in the population. The reasoning of tests goes like this: 1. Choose a statistic that measures the effect you are looking for. 2. Construct the sampling distribution that this statistic would have if the effect were not present in the population. 3. Locate the observed statistic on this distribution. A value in the main body of the distribution could easily occur just by chance. A value in the tail would rarely occur by chance and so is evidence that something other than chance is operating. The statement that the effect we seek is not present in the population is the null hypothesis, H0. Assuming the null hypothesis is true, the probability that we would observe a statistic value as extreme or more extreme than the one we did observe is the P-value. Figure 16.23 illustrates the idea of a P-value. 16.5 Significance Testing Using Permutation Tests When you complete this section, you will be able to Outline the steps needed for a permutation test for comparing two means. Outline the steps needed for a permutation test for a matched pairs study. Outline the steps needed for a permutation test for the relationship between two quantitative variables. tests of significance, p. 372 LOOK BACK (c) Give the 95% BCa confidence interval for the slope b1 of the population regression line. Compare this with the standard 95% confidence interval based on Normality, the bootstrap t interval, and the bootstrap percentile interval. Using the BCa interval as a standard, which of the other intervals are adequately accurate for practical use?