invest_3ed.pdf

# The above graphs show data that satisfy these

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The above graphs show data that satisfy these conditions nicely, and in this case we would consider the t -procedures from Investigation 5.10 to be valid. V E(Y at x )= E 0 + E 1 x Explanatory variable ( x ) Response variable ( y ) Explanatory variable ( x )

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Chance/Rossman, 2015 ISCAM III Investigation 5.13 394 Investigation 5.13: Cat Jumping (cont.) Reconsider the data from Investigation 5. 6 about factors that are related to a cat’s jumping ability ( CatJumping.txt ). (a) Use technology to determine the equation of the least squares line for predicting a cat’s takeoff velocity from its percentage of body fat. Record the equation of this line, using good statistical notation. (b) Produce a histogram and a normal probability plot of the residuals of this regression. x In R You can access the residuals by using > lm(price~sqft)\$residuals x In Minitab Run the regression again, but this time click the Storage button. In the Regression Storage window, check the Residuals box. Minitab will automatically create a column with all of the residual values calculated (RESI1). Does the normality condition appear to be satisfied? (c) Produce a graph of the residuals vs. the percentage of body fat variable. Does the equal variance condition appear to be met? Does the linearity condition appear to be met? Equal variance: Linearity: (d) Consider testing whether the sample data provide strong evidence that percentage of body fat has a negative association with takeoff velocity in the population. State the hypotheses to be tested, and report (from the output) the value of the appropriate test statistic and p-value. Summarize your conclusion. (e) Confirm that the t -test statistic for the hypotheses in (d) is equal to b 1 / SE ( b 1 ). (f) Use a t -procedure and the values of b 1 and SE ( b 1 ) to produce (by hand) a 95% confidence interval for the population slope β 1 .
Chance/Rossman, 2015 ISCAM III Investigation 5.13 395 (g) Interpret the confidence interval that you produced in (f). Be sure to interpret not only the interval itself but also what the slope coefficient means in this context. (h) Use the equation of the least squares line to predict the takeoff velocity for a cat with 25 percent body fat. Then do the same for a cat with 50 percent body fat. Predicted takeoff velocity with 25% body fat: Predicted takeoff velocity with 50% body fat: (i) Which of the two predictions in (h) do you feel more comfortable with? Which do you suspect would be more precise ? Explain. Rather than only report one number as our prediction, we would like specify a confidence interval that indicates our “accurate” or precise we believe our prediction to be . The following procedure is valid if the basic regression model conditions are met. Technology Detour Prediction Intervals In R Use the predict command, passing in the variables for the linear model but also a “new data” data frame for the value(s) you want predictions for (but giving them the same name as your explanatory variable).

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