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Sample Problems in Statistics and Research

## Problem 1: Chili Cook-Off Competition Sample Size

Prepare the solution for your choice for five of the following six problems. Clearly indicate each problem you have chosen to solve, and circle your final responses for each. Show all work. Submit your work to the designated file under Course Materials in Blackboard. If using software, submit a printout of your work. On the printout, clearly label the steps in your solution and circle your final responses. This assignment is due December 5. No extensions, no exceptions.

Imagine you are in charge of planning a chili cook-off competition among several cooking teams, sponsored by ASI here at CSUEB. You want to make sure that there are enough plates of chili for all pre-paid tasters. CSUEB has a total student body of about 13,000. You have the following standards you must meet: a confidence level of 99%, and an error of no more than four ounces per cooking team. Last year’s competition had a standard deviation of 30 ounces for the amount of chili cooked.
a. What is the necessary minimum sample size?
b. How is the sample size impacted if we change our confidence level to 90%?
c. How is the sample size impacted if we change the error requirement to no more than two ounces per cooking team?
d. Explain how determining sample size for the campus chili competition represents statistical,financial and organizational challenges.

Red Widget Strategies wants to improve its customer service by being more efficient in how phone calls are processed. Toward that end, Red Widget does a statistical analysis of 1,000 outbound phone calls. The length of the calls is normally distributed, with a population mean = 240 seconds, and a population standard deviation = 40 seconds.
a. What is the probability that a particular call lasted less than 180 seconds?
b. What is the probability that a particular call lasted between 180 and 300 seconds?
c. What is the probability that a call lasted between 110 and 180 seconds?
d. What is the probability that a call lasted more than 300 seconds?

A confidential labor organizers’ poll was recently conducted in a major discount retail chain. Based on a random sample of 1000 rank and file employees, the researchers classified employees by gender and asked them if they planned to support unionization, by responding “Union” or “No Union.” Results are shown in the following table.
Union No Union Total
Male 200 250 450
Female 300 250 550
Total 500 500 1000

a. State the null and alternative hypotheses.
b. Calculate the expected counts.
c. Calculate the value of the chi-square test statistic; assume a 95% confidence level.
d. Develop a conclusion. What can be said about the relationship between employee gender and preference for union representation?
e. Comment on the strengths and drawbacks of using the chi square test statistic.

Both employers and insurers are on the lookout for ways to control the costs of health insurance. Toward that end, an insurer-led study of workers in the financial services industry concerns the average fat loss between those who only diet, and those who only exercise. Following are the sample results from study participants on the amount of fat lost after one year:

Group 1: Diet Only
Sample mean: 5.9 kg
Sample standard deviation: 4.1 kg
Level of significance: .05
Number of participants: 42
Group 2: Exercise Only
Sample mean: 4.1 kg
Sample standard deviation: 3.7 kg
Level of significance: .05
Number of participants: 47

Research question: Is there a significant difference in average fat lost between these two methods?
a. Determine the null and alternate hypotheses.
b. Calculate the appropriate test statistic.
c. Find the p-value.
d. What do you conclude? Be specific.
5. Box Office Boffo!

A research project is organized at Red Widget Entertainment Studios to explore the relationship years of service and box office performance of star contract employees. The management objective is to project box office performance (rating scale = 0 to 6) based upon years of service.Employee Years of Service Performance Rating
Natalie 1 6
Meryl 20 5
Matt 8 5
George 2 2
Cate 1 2
Gwynneth 15 4