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Chapter 14: Problem 19

The article "Impacts of On-Campus and Off-Campus Work on First-Year Cognitive Outcomes" (Journal of College Student Development \([1994]: 364-370\) ) reported on a study in which \(y=\) spring math comprehension score was regressed against \(x_{1}=\) previous fall test score, \(x_{2}=\) previous fall academic motivation, \(x_{3}=\) age, \(x_{4}=\) number of credit hours, \(x_{5}=\) residence \((1\) if on campus, 0 otherwise), \(x_{6}=\) hours worked on campus, and \(x_{7}=\) hours worked off campus. The sample size was \(n=210\), and \(R^{2}=.543\). Test to see whether there is a useful linear relationship between \(y\) and at least one of the predictors.

Short Answer

Expert verified
A hypothesis test should be conducted with the F-statistic calculated using provided values. The F-statistic needs to be compared to a critical value from the F-distribution table matching corresponding degrees of freedom. The decision on whether or not to reject the null hypothesis will be based on the comparison of these two values.

Step by step solution

01

Identify Variables

First, recognize important variables: the coefficient of determination \(R^2 = .543\), the number of predictors \(k = 7\) (variables \(x_1\) to \(x_7\)), and the sample size \(n = 210\).
02

Calculate the F-Statistic

Use the values obtained in Step 1 to calculate the F-statistic using the formula: \[ F = \frac{(R^2 / k)}{((1-R^2) / (n-k-1))} \] Substituting the given values, the F-statistic value is calculated.
03

Calculate the Critical Value and Make a Decision

Next, determine the critical value based on the F-distribution table with degrees of freedom \(df1 = k\) and \(df2 = n-k-1\). If the computed F-statistic value is larger than the critical value, reject the null hypothesis. This shows that at least one predictor is useful in the linear relationship with \(y\). Otherwise, if the F-statistic value is less than or equal to the critical value, do not reject the null hypothesis, indicating that there is not enough evidence to prove the usefulness of at least one predictor.

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