The annual dividend rates for a random sample of 16 companies in three different industries,utilities,banking,and insurance,were recorded. The ANOVA comparing the mean annual dividend rate among three industries rejected the null hypothesis that the dividend rates were equal. The mean square error (MSE)was 3.36. The following table summarized the results: When comparing the mean annual dividend rate for companies in the utilities and insurance industries,which of the following 95% confidence intervals can be constructed?
A)5.78 ± 2.160 * 2.40
B)5.78 ± 2.120 * 2.40
C)5.78 ± 2.160 * 1.11
D)5.78 ± 2.120 * 1.11

Question

The annual dividend rates for a random sample of 16 companies in three different industries,utilities,banking,and insurance,were recorded. The ANOVA comparing the mean annual dividend rate among three industries rejected the null hypothesis that the dividend rates were equal. The mean square error (MSE)was 3.36. The following table summarized the results:  When comparing the mean annual dividend rate for companies in the utilities and insurance industries,which of the following 95% confidence intervals can be constructed?
A)5.78 ± 2.160 * 2.40 
B)5.78 ± 2.120 * 2.40 
C)5.78 ± 2.160 * 1.11 
D)5.78 ± 2.120 * 1.11

Quizplus · Accepted Answer

Since we are comparing the mean dividend rate between utilities and insurance, we only compare two industries, so the degrees of freedom for the error term is (16-3)=13. From the table, we can find the F-statistic for the utilities vs insurance comparison is 5.76. Using the MSE given in the problem, we can calculate the standard error of the mean difference between the two industries: sqrt(3.36/16 * (1/8 + 1/8)) = 0.655. The t-value for a 95% confidence interval with 13 degrees of freedom is 2.160. Therefore, the 95% confidence interval for the difference in mean annual dividend rates between the utilities and insurance industries is calculated as:
5.78 (utility mean) - 6.89 (insurance mean) ± 2.160 * 0.655 = 5.78 - 6.89 ± 1.413
= (-2.023, -0.747)
Hence, the answer is C.

The Annual Dividend Rates for a Random Sample of 16