Question

In Exercises 9.11 to \(9.14,\) test the correlation, as indicated. Show all details of the test. Test for a positive correlation; \(r=0.35 ; n=30\).

Short answer

To conclude if there's a positive correlation, we need to compare the calculated t-value (using the provided correlation coefficient and sample size) to the critical t-value for a one-tailed test at a significance level of 0.05. We reject the null hypothesis of no positive correlation if the calculated t-value exceeds the critical t-value, indicating a significant positive correlation. Otherwise, we fail to reject the null hypothesis, signifying no significant positive correlation.

Step-by-step solution

State the Hypotheses

Firstly, we identify our null and alternative hypotheses. The null hypothesis ( \( H_0 \) ) posits no positive correlation, so it would state that the correlation coefficient is equal to 0. The alternative hypothesis ( \( H_1 \) ) claims a positive correlation, therefore, it would state that the correlation coefficient is greater than 0. Expressed mathematically, this would be: \( H_0: r = 0 \) and \( H_1: r > 0 \).

Calculate the Test Statistic

We use the given correlation coefficient ( \( r = 0.35 \) ) and sample size ( \( n = 30 \) ), along with the equation for the t-test for correlation coefficients to calculate our test statistic. This equation is \( t = r \sqrt{\frac{n-2}{1-r^2}} \). Substituting our given values into this equation yields \( t = 0.35 \sqrt{\frac{30-2}{1-0.35^2}} \).

Compare to Critical Value

Next, we find the critical t-value for a one-tailed test (since we are only interested in determining whether the correlation is greater than 0) at a significance level of 0.05 with \( n-2 \) degrees of freedom. Using a t-table, we find the one-tailed critical t-value to be approximately 1.699. Then, we compare our calculated t-value to this critical value.

State the Conclusion

If our calculated t-value is greater than the critical t-value, we reject the null hypothesis in favor of the alternative hypothesis. This would indicate a significant positive correlation. If our calculated t-value is less than or equal to the critical t-value, we do not reject the null hypothesis. This would indicate that the correlation is not significantly different from 0.