Question

Q1 and Q3 are quite straightforward. You only need to state the null and alternative hypothesis. No calculations involved

University X has 20,000 students, of which 9000 are men, and 11,000 are women. You collected independent simple random samples from men and women, and recorded how much money each respondent spent on cellular telephone service in a month.

You got the following results:

1. Men: Sample size = 100, sample mean = $40, sample standard deviation = $20
2. Women: Sample size = 100, sample mean = $55, sample standard deviation = $15

Please answer the following three questions.


Q1: You want to test the null hypothesis that the total expenditure on cellular telephone service by all the 11000 women, combined, in a month does not exceed $500,000. Please write down the correct expression of the corresponding null hypothesis. Discuss the potential Type I and Type II error.


Q2: At a 99% level of confidence, test the above null hypothesis, that the total expenditure on cellular telephone service by all the 11000 women, combined, in a month does not exceed $500,000. Note: the corresponding Z statistic for 99% level of confidence is 2.33. (Show details of work)


Q3: You want to test the null hypothesis that on the average, the monthly expenditure on cellular telephone service by a woman does not exceed the monthly expenditure on cellular telephone service by a man by more than $10. Please write down the correct expression of the corresponding null hypothesis (You do not need to test this one).

Answer 1

Q1

Null Hypothesis : Total Expenditure = $500,000

Alternate Hypothesis : Total Expenditure > $500,000

Q2  

1. Set hypothesis

2. Alpha = 0.01 (significance level)

3. n = 100

Sample Standard Deviation = $15

Since sample size is greater than 30 , we can assume the distribution as normal and we can compute z statistic.

4. Z statistic = Sample Mean - Population Mean / standard error

Standard error = Standard Deviation /

= 15 / 10

= 1.5

Population Mean = $500,000/ 11000 = $45.45

Z statistic = $55- $ 45.45 / 1.5

= 6.3 (APPROX)

5. Critical Value = 2.33

6. Value of test statistic is greater than critical value

7. We can reject the null hypothesis.

Q3

Null hypothesis :  monthly expenditure of women = monthly expenditure of men

Alternate hypothesis : monthly expenditure of women = monthly expenditure of men + $10