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 28 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 = 13.9. A random sample of 36 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 = 10.1. Assume that σ₁ = 4.6 and σ₂ = 4.0.

(a) 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.

μ

p₁ – p₂  

  μ₁ – μ₂

p

(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 99% 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 99% 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?

Because the interval contains only positive numbers, we can say that the profit as a percentage of stockholder equity is higher for retail stocks.

Because the interval contains both positive and negative numbers, we can not say that the profit as a percentage of stockholder equity is higher for retail stocks.  

  We can not make any conclusions using this confidence interval.

Because the interval contains only negative numbers, we can say that the profit as a percentage of stockholder equity is higher for utility stocks.

Answer 1

a)
μ1 – μ2

b)

Pooled Variance
sp = sqrt(s1^2/n1 + s2^2/n2)
sp = sqrt(21.16/28 + 16/36)
sp = 1.0955

Given CI level is 0.99, hence α = 1 - 0.99 = 0.01                  
α/2 = 0.01/2 = 0.005, zc = z(α/2, df) = 2.58                  
                  
Margin of Error                  
ME = zc * sp                  
ME = 2.58 * 1.0955                  
ME = 2.826                  
                  
CI = (x1bar - x2bar - tc * sp , x1bar - x2bar + tc * sp)                  
CI = (13.9 - 20.1 - 2.58 * 1.0955 , 13.9 - 20.1 - 2.58 * 1.0955                  
CI = (-9.0 , -3.4)

c)

Because the interval contains only negative numbers, we can say that the profit as a percentage of stockholder equity is higher for utility stocks.