Question

It seems plausible that higher rent for retail space could be justified only by a higher level of sales. A random sample of \(n=53\) specialty stores in a chain was selected, and the values of \(x=\) annual dollar rent per square foot and \(y=\) annual dollar sales per square foot were determined, resulting in \(r=.37\) ("Association of Shopping Center Anchors with Performance of a Nonanchor Specialty Chain Store," Journal of Retailing [1985]: \(61-74\) ). Carry out a test at significance level \(.05\) to see whether there is in fact a positive linear association between \(x\) and \(y\) in the population of all such stores.

Short answer

To provide a definite answer, actual calculations will be needed in Step 2 and Step 3. If the calculated test statistic falls into the critical region, the conclusion is that there is a significant positive linear association between rent and sales per square foot. If not, then there is not a significant correlation.

Step-by-step solution

State the Null Hypothesis and Alternative Hypothesis

The null hypothesis (\(H_0\)) is that there is no linear association in the population, so the population correlation coefficient (\(\rho\)) is 0. The alternative hypothesis (\(H_a\)) is that there is a positive linear association in the population, so \(\rho > 0\). Therefore, \(H_0: \rho = 0\) and \(H_a: \rho > 0\).

Calculate the Test Statistic

The test statistic (t) for the correlation coefficient is given by the formula \(t = r\sqrt{(n-2)/(1-r^2)}\), where \(r\) is the sample correlation coefficient and \(n\) is the sample size. Plugging in the given values, \(t = 0.37\sqrt{(53-2)/(1-.37^2)}\). Calculate this value to obtain the test statistic.

Determine the Critical Region

The critical region for a one-tailed test at a significance level of \(.05\) with \(df = n - 2 = 51\) degrees of freedom is the set of t values greater than the critical t value. Using a t-distribution table or calculator, find the critical t value corresponding to \(.05\) level of significance and \(51\) degrees of freedom. Any test statistic greater than this critical value will fall into the critical (rejection) region.

Make a Decision

If the calculated test statistic from Step 2 falls into the critical region determined in Step 3, reject the null hypothesis. This would suggest that there is a significant positive linear association between rent and sales per square foot. If the calculated test statistic does not fall into the critical region, do not reject the null hypothesis. This would suggest that there is not a significant correlation between the two variables.