# Confidence Interval & Normal Probability

Q-3a. A stationery store wants to estimate the mean retail value of greeting cards purchased by its customers when they visit the store. A random sample of 15 customers indicates a mean amount of $2.55 with a standard deviation of $0.44 per customer per visit. Assuming a normal distribution, construct a 95% confidence interval estimate of the mean retail value of greeting cards purchased by its customers.

Q-3b. A paper manufacturer has a production process that operates continuously throughout an entire production shift. The paper is expected to have a mean length of 11 inches, and the standard deviation of the length is 0.02 inches. At periodic intervals, a sample is selected to determine whether the mean paper length is still equal to 11 inches or whether something has gone wrong in the production process to change the length of the paper produced. You select a random sample of 100 sheets, and the mean paper length is 10.998 inches. Construct a 95% confidence interval estimate for the population mean paper length.

Q-3c. A survey of 705 workers asked how much they used the Internet at work. 423 said they used it within limits, and 183 said that they did not use the Internet at work. Construct a 95% confidence interval estimate for the proportion of all workers who use the Internet within limits.

Q-3d. You are the designer for your company's web site. You have data to indicate that the mean download time for the homepage is 7 seconds and that the standard deviation of download time is 2.0 seconds. If we assume that the download times are normally distributed, what percent of users will wait between 5 and 10 seconds for the homepage to download?

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#### Solution Summary

The solution provides step by step method for the calculation of confidence interval and probability using the Z score. Formula for the calculation and Interpretations of the results are also included.

Statistics: Chapter 7 and 8 problems

See attached file.

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).