For a sample of 40 large U.S. cities, the correlation between the mean number of square feet per office worker and the mean monthly rental rate in the central business district is -0.363. At the .05 significance level, can we conclude that there is a negative association between the two variables.
Question
For a sample of 40 large U.S. cities, the correlation between the mean number of square feet per office worker and the mean monthly rental rate in the central business district is -0.363. At the .05 significance level, can we conclude that there is a negative association between the two variables.
Solution
To answer this question, we need to conduct a hypothesis test for correlation. Here are the steps:
Step 1: State the hypotheses. The null hypothesis is that there is no correlation between the mean number of square feet per office worker and the mean monthly rental rate in the central business district (ρ = 0). The alternative hypothesis is that there is a negative correlation between the two variables (ρ < 0).
Step 2: Determine the test statistic. The test statistic for a correlation is a t-score, which is calculated using the formula: t = r * sqrt[(n-2)/(1-r^2)], where r is the correlation coefficient and n is the sample size. In this case, r = -0.363 and n = 40.
Step 3: Calculate the test statistic. Substituting the given values into the formula, we get: t = -0.363 * sqrt[(40-2)/(1-(-0.363)^2)] = -2.41 (approx).
Step 4: Determine the critical value. For a one-tailed test at the 0.05 significance level with 38 degrees of freedom (n-2), the critical value from the t-distribution table is approximately -1.686.
Step 5: Make a decision. Since the calculated t-score (-2.41) is less than the critical value (-1.686), we reject the null hypothesis.
Therefore, at the 0.05 significance level, we can conclude that there is a negative association between the mean number of square feet per office worker and the mean monthly rental rate in the central business district.
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