Question

A sample of \(n=353\) college faculty members was obtained, and the values of \(x=\) teaching evaluation index and \(y=\) annual raise were determined ("Determination of Faculty Pay: An Agency Theory Perspective," Academy of Management Journal [1992]: 921-955). The resulting value of \(r\) was .11. Does there appear to be a linear association between these variables in the population from which the sample was selected? Carry out a test of hypothesis using a significance level of \(.05\). Does the conclusion surprise you? Explain.

Short answer

Without calculating the exact values, the general process involves comparing the test statistic and the critical value. Depending on the comparison's result, a conclusion can be drawn about whether there is a significant linear relationship between the teaching evaluation index and the annual raise. The detailed surprising nature of the conclusion requires the context of the situation.

Step-by-step solution

Identify the Hypotheses

Firstly, it should be pointed out the null hypothesis and the alternative hypothesis. The null hypothesis (\(H_0\)) is that there is no linear association between the teaching evaluation index and the annual raise. Thus, the population correlation (\(ρ\)) is zero. The alternative hypothesis (\(H_A\)) is that there is a linear association, meaning \(ρ\) is not zero.Thus, these can be written as:\(H_0 : ρ = 0\)\(H_A : ρ ≠ 0\)

Calculate the Test Statistic

The formula used to calculate the test statistic for a correlation coefficient is:\( t = r \sqrt{\frac{n-2}{1-r^2}} \)where \( r \) is the sample correlation, and \( n \) is the sample size.Substitute given values ( \( r = .11 \), \( n = 353 \)) to the formula, which gives t.

Determine the Critical Value

To make a decision about the null hypothesis, compare the absolute value of test statistic with the critical value for the given significance level (\(\alpha = .05\)) and degrees of freedom (\(df = n - 2 = 351\)). The critical value can be found from the t-distribution table.

Make the Decision

If the absolute value of the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis. This suggests whether or not there is a significant linear association between the variables in the population.

Interpret the Results

Based on the decision in the previous step, provide an answer to the problem. If the null hypothesis was rejected, it means that there is a significant linear relationship between the teaching evaluation index and the annual raise. Otherwise, there is no evidence to suggest a significant relationship.