Does a large value of R2 always imply that at least one of

Chapter 11, Problem 87E

(choose chapter or problem)

Does a large value of \(R^{2}\) always imply that at least one of the independent variables should be retained in the regression model? Does a small value of \(R^{2}\) always indicate that none of the independent variables are useful for prediction of the response?

a Suppose that a model with k = 4 independent variables is fit using n = 7 data points and that \(R^{2}=.9\). How many numerator and denominator degrees of freedom are associated with the F statistic for testing \(H_{0}=\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0\)? Use the result in Exercise 11.84(a) to compute the value of the appropriate F statistic. Can H0 be rejected at the α = .10 significance level?

b Refer to part (a). What do you observe about the relative sizes of n and k? What impact does this have on the value of F?

c A model with k = 3 independent variables is fit to n = 44 data points resulting in \(R^{2}=.15\). How many numerator and denominator degrees of freedom are associated with the F statistic for testing \(H_{0}=\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0\)? Use the result in Exercise 11.84(a) to compute the value of the appropriate F statistic. Can H0 be rejected at the \(\alpha=.10\) significance level?

d Refer to part (c). What do you observe about the relative sizes of n and k? What impact does this have on the value of F?

Equation transcription:

Text transcription:

R^{2}

R^{2}=.9

H{0}=beta{1}=beta{2}=beta{3}=beta{4}=0

alpha=.10

R^{2}=.15