Question

A sample of \(n=10,000(x, y)\) pairs resulted in \(r=\) .022. Test \(H_{0}: \rho=0\) versus \(H_{a}: \rho \neq 0\) at significance level .05. Is the result statistically significant? Comment on the practical significance of your analysis.

Short answer

To arrive at a short answer, the value of Z from the calculation in Step 1 is needed for comparison with the critical value. If |Z| > 1.96, the result is statistically significant and reject the null hypothesis. If |Z| ≤ 1.96, do not reject the null hypothesis. In terms of practical significance, considering the weak correlation of 0.022, there may not be any practical significance even if the result is statistically significant.

Step-by-step solution

Calculate the test statistic

The test statistic Z for the correlation coefficient is determined by the formula \(Z = r \sqrt{(n - 2) / (1 - r^2)}\). Substituting the given values, we have \(Z = 0.022 \sqrt{(10,000 - 2) / (1 - 0.022^2)}\). Compute this expression to obtain the numerical value of Z.

Determine the critical value

The problem gives a significance level of 0.05. Since the problem does not specify the type of test (one-tailed or two-tailed), we assume a two-tailed test because the alternative hypothesis \(H_{a}: \rho \neq 0\) doesn't specify a direction. In this case, we will reject the null hypothesis if our test statistic falls in the top 2.5% or bottom 2.5% of the standard normal distribution. That’s because under the null hypothesis, we assume the correlation coefficient will follow a standard normal distribution. The critical value for a two-tailed test at 0.05 level is approximately 1.96.

Make a decision

After obtaining the value of Z from Step 1, compare this with the critical value from Step 2. If the absolute value of Z is greater than the critical value, reject the null hypothesis in favor of the alternative hypothesis. Otherwise, do not reject the null hypothesis.

Consider practical significance

If the result is statistically significant, discuss its practical implications. Keep in mind that even though a result might be statistically significant, it might not have practical significance, especially if the correlation is very weak. In practical terms, a correlation of 0.022 is relatively weak, which means the variables are barely if at all, related.