Statistics (Math 141)
Statistics (Math 141)
Homework: B6
Instructions
Please complete all problems. Unless otherwise noted, each problem is worth 5 points. To obtain full credit, you must use the methods specified in the course text. Please, format your work as explained in chapter 2 of the Frequently Asked Questions (FAQ) document. When doing calculations, such as the variance, standard deviation, or the correlation coefficient, use technology wherever possible. There is no need to do calculations by hand unless I direct you to do so.
Rounding. You should round your z-scores to twodecimal places when using a normal distribution table and the areas under the normal curve to decimal places. When using the t-distribution or doing other calculations, round to 3 decimal places unless otherwise directed. Never round until the last step of your calculations (Note: see FAQ 4.2).
Level of confidence, c. Many of the calculations in this unit start after the value of c is identified. For this reason, you should take care to thoroughly understand how c is defined. You need to understand the relationship between c and critical values.
Finding confidence intervals using a calculator.In addition to knowing how to hand calculate confidence intervals, you should check out the ZInterval() and TInterval() functions in the STAT TESTS submenu of the TI83/84.
Expressing confidence intervals [See FAQ 4.8].You should express a confidence interval using the interval notation (a,b)(a,b), where a is the lower bound and b is the upper bound. The same notation should be used whether you are doing a confidence interval for the normal distribution, the t-distribution, or for a proportion. It is not necessary to provide a number line representation for these intervals. Unless otherwise directed, round the values you calculate for a and b in the confidence interval to three decimal places.
Deciding between the Normal distribution and the t-distribution. You will need to know how to decide whether to use the normal distribution or the t-distribution, so you should carefully review and understand the criteria in chapter 6 of the text.
Sample size calculations. Many students have difficulty doing these calculations. In most cases, the errors are caused by a failure to force the calculator to follow the order of operations correctly for the problem. I suggest that you carefully review the examples pertaining to these calculations that are given in the text. Do the calculations and ensure that you get the same answer given by the author. Then follow the same procedure to do the calculations for the homework. (Note: see this topic in chapter 4 of the FAQ document.)
Assignment Problems
Important! Assume that all samples mentioned in this assignment are randomly selected.
Section 6.1
1. When estimating a population parameter from sample statistics, rather than estimating a specific value, what is used?
2. If μ=12.74μ=12.74 and ¯¯¯x=10.87x¯=10.87. What is the sampling error?
3. Let c=0.90c=0.90, σ=9.6σ=9.6, and n=34n=34. What is the margin of error, E?
Problems 4-5. Use the confidence interval for the population mean, (54.32,72.18)(54.32,72.18), to answer the questions.
4. What is the margin of error, EE?
5. What is the sample mean, ¯¯¯xx¯?
6. Minimum sample size. Let c=0.95c=0.95, σ=10.4σ=10.4, and E=2.1E=2.1. What is the minimum sample size, nn, needed to estimate μμ? (see FAQ chapter 4.)
7. Repair Costs: Refrigerators. In a random sample of 50 refrigerators, the mean repair cost was $194.00. The population standard deviation, σ=$22.50σ=$22.50. Construct a 90% confidence interval for the population mean repair cost.
Section 6.2
Problems 8-10. Assume that σσ is not known, the sample is random, and the population is normally distributed.
8. Let c=0.99c=0.99 and n=33n=33. Find the critical value, tctc.
9. Let c=0.95c=0.95, s=6.9s=6.9, and n=40n=40. Find the margin of error, EE.
10. Let c=0.98c=0.98, ¯¯¯x=53.7x¯=53.7, s=6.4s=6.4, and n=32n=32. Construct a confidence interval for the population mean μμ using the t-distribution.
Problems 11-12. Repair Costs: Computers. In a random sample of 20 computers, the mean repair cost was $142.40 and the standard deviation was $33.65. Assume the repair cost is normally distributed.
11. Use a t-distribution to construct a 99% confidence interval for the population mean μμ.
12. What is the margin of error, EE?
Problems 13-15. Salaries. The monthly incomes for 14 randomly selected people, each with a bachelor’s degree were gathered. Assume that the population of the monthly incomes is normally distributed. Use the data in Table 1 to answer the questions.
Table 1. The monthly incomes of 14 randomly selected people with a bachelor’s degree
Salaries (Dollars)
6399.84 7072.61 5847.37 6746.40 7344.15 7091.05 6312.74
6107.86 7521.89 6540.86 7063.97 7045.74 7637.07 8049.48
13. What is the sample mean, ¯¯¯xx¯?
14. What is the sample standard deviation, ss?
15. Construct a 90% confidence interval for the population mean μμ.
Section 6.3
Problems 16-17. Let pp be the population proportion. In a survey of 5218 U.S. adults, 3642 say they have contributed to a charity in the past 12 months. Round results to 3 decimal places.
16. What is the point estimate, ^pp^, for pp?
17. What is the point estimate, ^qq^, for qq?
Problems 18-19. Let pp be the population proportion. In a survey of 2248 U.S. adults, 679 say they would go on vacation to Europe if cost did not matter. Round results to 3 decimal places.
What is the point estimate, ^pp^, for pp?
What is the point estimate, ^qq^, for qq?
Problems 20-22. In a survey of 5350 U.S. adults, 427 say they have traveled out of country in the past 12 months. Round your results to 3 decimal places.
20. Construct the 95% confidence interval for the population proportion pp.
21. Construct the 99% confidence interval for the population proportion pp.
22. Which of the two intervals you found is narrower? (Suggestion: plot the interval boundaries on a number line.)
Problems 23-25. Online Service Usage. You are a travel agent and wish to estimate, with 92% confidence, the proportion of vacationers who use an online service or the internet to make travel reservations. Your estimate must be accurate to within 5% of the population proportion. (Note: Do not use the c value for 90% or 95% to approximate the solution. Use the characteristics of c to find the best zczc value for a 92% level of confidence. See FAQ 4.11.)
23. Assume that no preliminary estimate is available. Find the minimum sample size needed.
24. Find the minimum sample size needed, using a prior study that found that 40% of the respondents said they used an online service or the internet to make travel reservations.
25. What do you conclude about the effects of having a preliminary estimate based on your results found in questions 23 and 24?
Statistics (Math 141)
