How profitable are different sectors of the stock market? One way to answer such a question is to examine profit as a percentage of stockholder equity. A random sample of 32 retail stocks such as Toys “Я” Us, Best Buy, and Gap was studied for x1, profit as a percentage of stockholder equity. The result was 1 = 13.7. A random sample of 34 utility (gas and electric) stocks such as Boston Edison, Wisconsin Energy, and Texas Utilities was studied for x2, profit as a percentage of stockholder equity. The result was 2 = 10.1. Assume σ1 = 4.1 and σ2 = 2.7.
(a) Let μ1 represent the population mean profit as a percentage of stockholder equity for retail stocks, and let μ2 represent the population mean profit as a percentage of stockholder equity for utility stocks. Find a 95% confidence interval for μ1 - μ2.
(b) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? All negative? Of different signs? At the 95% level of confidence, does it appear that the profit as a percentage of stockholder equity for retail stocks is higher than that for utility stocks?
(c) Test the claim that the profit as a percentage of stockholder equity for retail stocks is higher than that for utility stocks. Use α = 0.01.
For each hypothesis test, please provide the following information:
(i) What is the level of significance? State the null and alternate hypotheses.
(ii) What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic?
(iii) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.
(iv) Based on your answers in parts (i) – (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
(v) Interpret your conclusion in the context of the application.