Course OPRE: 506

Course Material:

Book: Title: Statistics for Managers Using Microsoft Excel

Authors: Levine, David M., Stephan, David F. and Szabat, Kathryn A.

Edition: 9^th Year: 2021

Publisher: Pearson

Instructions must follow:

- Do not use any artificial intelligence tools (ChatGPT or any other similar software) to answer problems.

- Include all details and steps performed to find your answers.

- Must do STEP by Step Solutions properly.

Please answer the following questions giving all details.

1. The manager of a grocery store claims that the mean shopping time in her store is 20 minutes. In a sample of 100 shoppers at that grocery store, the mean shopping time was 21.50 minutes with a standard deviation of 5.50 minutes. Using the 0.05 level of significance, is there evidence that the mean shopping time at the grocery store is different from the claimed value of 20 minutes?

a. State the null and alternative hypothesis for this problem.

b. Determine the sample test statistic value for this problem.

c. Determine the critical test statistic value for this problem.

d. What is your statistical decision for this problem?

2. The operations manager at a light bulb factory wants to determine whether there is any difference in the mean life expectancy of bulbs manufactured on two different types of machines. A random sample of 25 light bulbs from machine I indicates a sample mean of 400 hours and a sample standard deviation of 100 hours. A similar sample of 25 light bulbs from machine II indicates a sample mean of 410 hours and a sample standard deviation of 110 hours. For this problem:

a. What is the value of the pooled-variance t_STAT test statistic for testing H₀: ₁ = ₂?

b. In finding the critical value, how many degrees of freedom are there?

c. Using the level of significance = 0.01, what is the critical value for a one-tail test of the hypothesis H₀: ₁ < ₂ against the alternative H₁: ₁ > ₂?

d. What is your statistical decision?

3. To test the effectiveness of a business school preparation course, 10 students took a general business test before and after the course. The results are given below.

Student	Exam Score Before the Course	Exam Score After the Course
1	640	790
2	700	790
3	920	1,020
4	760	730
5	460	570
6	850	880
7	830	790
8	640	630
9	590	605
10	680	780

a. Use an appropriate hypothesis test, at 0.05 level of significance, to determine whether there is evidence of a difference between before and after scores of the students.

b. What assumption is necessary to perform the hypothesis test in (a)?

c. Construct a 95% confidence interval estimate of the mean difference in before and after scores. Interpret the interval. What is your decision based on this confidence interval estimate?

4. The following information is available for two samples selected from independent normally distributed populations.

Population A: n₁ = 25, S₁ = 7.5

Population B: n₂ = 20, S₂ = 6.4

a. At the 0.05 level of significance, is there evidence of a difference between ₁² and ₂²?

b. Suppose that you want to perform a one-tail test. At the 0.05 level of significance, what is the upper-tail critical value of F to determine whether there is evidence that? What is your statistical decision?