How profitable are different sectors of the stock market? One way to answer such

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 32retail 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 = 14.2. 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 = 10.8. Assume that 1 = 3.6 and 2 = 2.8.

(a) Categorize the problem below according to parameter being estimated, proportion p, mean , difference of means 1 – 2, or difference of proportions p1 – p2. Then solve the problem.

1.) 1 – 2

2.) p1 – p2

3.) p

4.)

(b) Let 1 represent the population mean profit as a percentage of stockholder equity for retail stocks, and let 2 represent the population mean profit as a percentage of stockholder equity for utility stocks. Find a 95% confidence interval for 1 – 2. (Use 1 decimal place.)

(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?

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

2.) 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.    

3.) We can not make any conclusions using this confidence interval.

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

Explanation / Answer

Q1.
1.) 1 – 2

Q2.
TRADITIONAL METHOD
given that,
mean(x)=14.2
standard deviation , 1 =3.6
population size(n1)=32
y(mean)=10.8
standard deviation, 2 =2.8
population size(n2)=36
I.
stanadard error = sqrt(s.d1^2/n1)+(s.d2^2/n2)
where,
sd1, sd2 = standard deviation of both
n1, n2 = sample size
stanadard error = sqrt((12.96/32)+(7.84/36))
= 0.7892
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 0.7892
= 1.5468
III.
CI = (x1-x2) ± margin of error
confidence interval = [ (14.2-10.8) ± 1.5468 ]
= [1.8532 , 4.9468]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
mean(x)=14.2
standard deviation , 1 =3.6
number(n1)=32
y(mean)=10.8
standard deviation, 2 =2.8
number(n2)=36
CI = x1 - x2 ± Z a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
where,
x1,x2 = mean of populations
sd1,sd2 = standard deviations
n1,n2 = size of both
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ ( 14.2-10.8) ±Z a/2 * Sqrt( 12.96/32+7.84/36)]
= [ (3.4) ± Z a/2 * Sqrt( 0.6228) ]
= [ (3.4) ± 1.96 * Sqrt( 0.6228) ]
= [1.8532 , 4.9468]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 95% sure that the interval [1.8532 , 4.9468] contains the difference between
true population mean U1 - U2
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the difference between
true population mean U1 - U2
3. Since this Cl does contain a zero we can conclude at 0.05 true mean
difference is zero

Q3.
1.) Because the interval contains only positive numbers, we can say that the profit as a percentage of stockholder equity is higher for retail stocks.

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