Question

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 25 retail stocks such as Toys 'R' Us, Best Buy, and Gap was studied for x₁, profit as a percentage of stockholder equity. The result was x₁ = 14.8. A random sample of 30 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 x₂ = 10.0. Assume that σ₁ = 4.9 and σ₂ = 2.3.

Categorize the problem below according to parameter being estimated, proportion p, mean μ, difference of means μ₁ – μ₂, or difference of proportions p₁ – p₂. Then solve the problem.

pμ₁ – μ₂    p₁ – p₂μ

(b) 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. (Use 1 decimal place.)

lower limit
upper limit

(c) 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?

Answer 1

The problem is categorized as difference of means.

We need to construct the 95% confidence interval for the difference between the population means μ1​−μ2​. The following information has been provided about each of the samples:

Sample Mean 1 =	14.8
Pop. Standard Deviation 1 (σ1​) =	4.9
Sample Size 1 (N1​) =	25
Sample Mean 2 =	10
Pop. Standard Deviation 2 (σ2​) =	2.3
Sample Size 2 (N2​) =	30

The critical value for α=0.05 is .

The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the difference between the population means 2.71 < μ1​−μ2 ​< 6.89, which indicates that we are 95% confident that the true difference between population means is contained by the interval (2.71,6.89) (2.7, 6.9).

The interval consists of all positive numbers.

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?

We need to perform a statistical test.

1 denotes the retail stocks and 2 denotes the utility stocks.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha : μ1 ​> μ2​

This corresponds to a right-tailed test, for which a z-test for two population means, with known population standard deviations, will be used.

(2) Rejection Region

The significance level is α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is R = { z : z > 1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = 1.45 ≤ zc ​= 1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0735, and since p = 0.0735 ≥ 0.05, it is concluded that the null hypothesis is not rejected.

At the 95% level of confidence, it does not appear that the profit as a percentage of stockholder equity for retail stocks is higher than that for utility stocks