Confidence Intervals & Hypothesis Testing

Newt Dimswitch owns a small construction company in Chesapeake, Virginia. He often uses more than one construction crew, and has had a difficult time maintaining records on profit levels, cost controls, and other variables that are necessary in making business decisions. Newt somehow feels that the use of one of his construction crews is not proving as profitable as other crews when sent out on a job. To make some kind of intelligent estimate of profit levels Newt collects cost data for the crew in question for the last 28 jobs it completed. Completion times, in hours, for the these jobs were 79, 83, 77, 81, 65, 92, 72, 71, 90, 79, 73, 70, 64, 91, 77, 80, 65, 85, 65, 89, 80, 65, 73, 78, 68, 88, 82, and 69. Newt must pay his crew a total of $112 per hour while it is on the job. This crew has bought in revenues on these jobs of $11200, 11700, 11500, 11100, 8300, 11000, 9900, 9400, 11500, 9200, 10200, 9100, 8400, 10300, 10900, 10900, 8300, 9700, 8600, 11200, 11200, 8600, 9200, 10900, 8700, 11100, 10400, 8400.

Now that Newt has these data, he isn’t sure what to do with them. He heard something about confidence intervals could be used to make estimates. Your job is to help poor Newt decide if this crew is profitable.

1. (5 points) MINITAB output. Enter the completion time data in one column. Use Time as your header. Select Stat > Basic Statistics > 1 – sample t... The variable is Time. Click OK. MINITAB will yield a 95% confidence interval for the mean job completion time for this crew.
2. (5 points) Determine the 95% confidence interval for cost of a job by this crew. (Hint: use interval from question 1) Show your calculations
3. (5 points) MINITAB output. Enter the revenue data in one column. Use Revenue as your header. Select Stat > Basic Statistics > 1 – sample t... The variable is Revenue. Click OK. MINITAB will yield a 95% confidence interval for the mean revenue for this crew.
4. (5 points) Now, is this crew making a profit? Determine the 95% confidence interval for profit. (Hint: use intervals from questions 2 & 3). Show your calculations

A group of business students conducted a survey on their campus to determine demand for a particular product, a protein supplement for Smoothies. As part of their initial steps they randomly sampled 113 students and obtained data that could be helpful in developing their marketing strategy. The responses to this survey are listed below.

Yes Yes Yes No Yes Yes No Yes Yes No Yes Yes Yes

Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes Yes

No No No Yes No No No No No No Yes Yes Yes

Yes No Yes No No Yes No Yes No Yes Yes No Yes

Yes Yes Yes No No Yes Yes Yes Yes Yes No Yes No

No Yes Yes Yes Yes No No Yes No Yes Yes Yes Yes

No No Yes Yes No Yes No Yes Yes No No No Yes

No Yes Yes Yes Yes No No Yes No Yes Yes Yes Yes

Yes Yes No Yes No Yes Yes Yes No

5. (5 points) MINITAB Output. Enter these data into a MINITAB worksheet all in one column. Use Smoothie as your header. Select Stat > Basic Statistics > 1-Proportion. Complete the dialog in order to obtain a 98% confidence interval.

6. (5 points) Based on the 98% confidence interval for students who would like protein supplements in their Smoothies, do you have sufficient evidence to conclude that more than 55% of students prefer a protein supplement for Smoothies? Explain.

The branch manager of a pet supply store wants to study characteristics of customers of his store. In particular, he decides to focus on two variables: the amount of money spent by customers and whether the customers own only one cat.

7. (5 points) If he wants to have 95% confidence of estimating the true population mean amount spent in his store within ± $1.50 and the standard deviation is assumed to be $10, what sample size is needed? Show your calculations

8. (5 points) If he wants to have 90% confidence of estimating the population proportion of customers who own only a cat to within ± 0.045, what sample size is needed? Show your calculations

The Haines Lumber Company makes plywood for the furniture industry. One product it makes is ¾ - inch oak veneer panels. It is very important that the panels conform to the ¾ - inch specification. Each hour, 5 panels are selected at random and measured. After 4 hours a total of 20 panels have been measured. The thickness measures are shown below.

Hour Panel 1 Panel 2 Panel 3 Panel 4 Panel 5

1 0.745 0.715 0.762 0.794 0.792

2 0.808 0.687 0.747 0.788 0.721

3 0.733 0.702 0.697 0.724 0.731

4 0.689 0.737 0.742 0.759 0.743

9. (5 points) What sampling technique was used to select this sample?

10. (5 points) State the null and alternative hypothesis.

11. (5 points) State the decision rule ( \(\alpha \) = 0.01).

12. (5 points) Calculate your test statistic. Show your calculations

13. (5 points) What is your decision regarding the null statement?

14. (5 points) Write a conclusion statement (interpret the statistical results in real world terms).

15. (5 points) Consider the results of the hypothesis test you conducted above. Which of the two types of errors could you have committed? Explain the ramification of committing such an error in the context of this scenario.

A study by the Centers for Disease Control (CDC) found that 23.3% of adults are smokers and that roughly 70% of those who do smoke indicated that they want to quit (Associated Press). CDC reported that, of people who smoked at some point in their lives, 50% have been able to kick the habit. Part of the study suggested that the success rate for quitting rose by education level. Assume that a sample of 100 college graduates who smoked at some point in their lives showed that 64 had been able to successfully stop smoking.

16. (5 points) State the hypotheses that can be used to determine whether the population of college graduates has a success rate higher than the overall population when it comes to breaking the smoking habit.

17. (5 points) Calculate the test statistic. Show your calculation

18. (5 points) Determine the p-value.

19. (5 points) What is your decision regarding the null statement if \(\alpha \) = 0.01?

20. (5 points) Write a conclusion statement.

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