Question

Test the correlation, as indicated. Show all details of the test. Test for evidence of a linear association; \(r=0.28 ; n=100\).

Short answer

The conclusion regarding the evidence of a linear association will depend on the calculated value of `t` from Step 3. If \( |t| > 1.984 \), there is evidence of a linear association. If \( |t| \leq 1.984 \), there is not enough evidence of a linear association.

Step-by-step solution

Compute test statistic

First, calculate the test statistic using the formula for testing a correlation coefficient, which is: \[ t = r \cdot \sqrt{ \frac{n-2}{1-r^{2}}} \]Substituting `r = 0.28` and `n = 100`, we get: \[ t \approx 0.28 \cdot \sqrt{ \frac{100-2}{1 - 0.28^2}} \]

Find the Critical Value

Next, find the critical value using the t-distribution table. With a significance level of 0.05 and degrees of freedom = `n-2 = 98`, the critical value for a two-tailed test (because we're looking for any association, positive or negative) is approximately 1.984.

Compare and Interpret

Once you've computed the test statistic and found the critical value, compare these values to make a decision. If the absolute value of the test statistic is greater than the critical value, reject the null hypothesis; there is a linear association. If the absolute value of the test statistic is less than or equal to the critical value, do not reject the null hypothesis; there is no sufficient evidence of a linear association. Compute the actual value of `t` from Step 1 and compare it to `1.984` to complete this step.