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# Normal Probability & Confidence Interval

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Chapter 7, problems 7.22, 7.23, 7.24, 7.25
Chapter 8, problems 8.32, 8.33, 8.61, 8.67

7.22. The US Census Bureau announces that the median sales price of new houses sold in March 2006 was \$224,200 while the mean sales price was \$ 279,100. Assume that the standard deviation of the prices is \$ 90,000.

a. If you select samples of n=2, describe the shape of the sample distribution of Xbar.
b. If you select samples of n=100, describe the shape of the sample distribution of Xbar.
c. If you select a random sample of n=100, what is the probability that the sample mean will be less than \$ 250,000?

7.23. Time spent using email per session is normally distributed with mu being 8 and sigma 2 minutes. If you select a random sample of 25 sessions,

a. What is the probability that the sample mean is between 7.8 and 8.2 minutes?
b. What is the probability that the sample mean is between 7.5 and 8.0 minutes?
c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 7.8 and 8.2 minutes?
d. Explain the difference in the results of (a) and (c).

7.24. The amount of time a bank teller spends with each customer has a population mean of 3.10 minutes and a standard deviation of 0.40 minutes. If you select a random sample of 16 customers,

a. What is the probability that the mean time spent per customer is at least 3 minutes?
b. There is an 85% chance that the sample mean is less than how many minutes?
c. What assumptions must you make in order to solve (a) and (b)?
d. If you select a random sample of 64 customers, there is an 85% chance that the sample mean is less than how many minutes?

7.25. The New York Times reported that the mean time to download the home page for the IRS was 0.8 seconds. Suppose that the download time was normally distributed with a standard deviation of 0.2 second. If you select a random sample of 30 download times,

a. What is the probability that the sample mean is less than 0.75 second?
b. What is the probability that the sample mean is between 0.70 and 0.90 second?
c. The probability is 80% that the sample mean is between what two values, symmetrically distributed around the population mean?
d. The probability is 90% that the sample mean is less than what value?

8.32. In a survey of 894 respondents with salaries below \$ 100,000 per year, 367 indicated that the primary reason for staying on their job was interesting job responsibilities.

a. Construct a 95% confidence interval for the proportion of all workers whose primary reason for staying on their job was interesting job responsibilities.
b. Interpret the interval constructed in (a).

8.33. A large number of companies are trying to reduce the cost of prescription drug benefits by requiring employees to purchase drugs through a mandatory mail-order program. In a survey of 600 employers, 126 indicated that they either had a mandatory mail order program in place or were adopting one by the end of 2004.

a. Construct a 95% confidence interval for the proportion of employers who had a mandatory mail-order program in place or were adopting one by the end of 2004.
b. Construct a 99% confidence interval for the proportion of employers who had a mandatory mail-order program in place or were adopting one by the end of 2004.
c. Interpret the interval constructed in (a).
d. Discuss the effect on the confidence interval estimate when you change the level of confidence.

8.61. When do you use the t distribution to develop the confidence interval estimate for the mean?

8.67. Companies are spending more time screening applicants than in the past. A study of 102 recruiters conducted by ExecuNet found that 77 did Internet research on candidates.

a. Construct a 95% confidence interval for the proportion of recruiters who do Internet research on candidates.
b. Based on (a), is it correct to conclude that more than 70% of recruiters do Internet research on candidates? Explain.
c. Suppose that the study uses a sample size of 400 recruiters and 302 did Internet research on candidates. Construct a 95% confidence interval estimate of the population proportion of recruiters who do Internet research on candidates.
d. Based on (c), is it correct to conclude that more than 70% of recruiters do Internet research on candidates? Explain.
e. Discuss the effect of sample size on your answers to (a) through (d).

https://brainmass.com/statistics/confidence-interval/normal-probability-confidence-interval-348026

#### Solution Summary

The solution provides step by step method for the calculation of normal probabilities and confidence interval for population proportion. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

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## Statistics: Sampling error, estimated population mean, confidence interval

21. Ch 8. What is sampling error? Could the value of the sampling error be zero? If it were zero, what would this mean?

22. Ch 8. List the reasons for sampling. Give an example of each reason for sampling.

34. Ch 8. Information from the American Institute of Insurance indicates the mean amount of life insurance per household in the United States is \$110,000. This distribution follows the normal distribution with a standard deviation of \$40,000.

a. If we select a random sample of 50 households, what is the standard error of the mean?
b. What is the expected shape of the distribution of the sample mean?
c. What is the likelihood of selecting a sample with a mean of at least \$112,000?
d. What is the likelihood of selecting a sample with a mean of more than \$100,000?
e. Find the likelihood of selecting a sample with a mean of more than \$100,000 but less than \$112,000.

32. Ch 9. A state meat inspector in Iowa has been given the assignment of estimating the mean net weight of packages of ground chuck labeled "3 pounds." Of course, he realizes that the weights cannot be precisely 3 pounds. A sample of 36 packages reveals the mean weight to be 3.01 pounds, with a standard deviation of 0.03 pounds.

A. What is the estimated population mean?
B. Determine a 95 percent confidence interval for the population mean.

34. Ch 9. A recent survey of 50 executives who were laid off from their previous position revealed it took a mean of 26 weeks for them to find another position. The standard deviation of the sample was 6.2 weeks. Construct a 95 percent confidence interval for the population mean. Is it reasonable that the population mean is 28 weeks? Justify your answer.

46. Ch 9 As a condition of employment, Fashion Industries applicants must pass a drug test. Of the last 220 applicants 14 failed the test. Develop a 99 percent confidence interval for the proportion of applicants that fail the test. Would it be reasonable to conclude that more than 10 percent of the applicants are now failing the test? In addition to the testing of applicants, Fashion Industries randomly tests its employees throughout the year. Last year in the 400 random tests conducted, 14 employees failed the test. Would it be reasonable to conclude that less than 5 percent of the employees are not able to pass the random drug test?

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