**The file P02_07.xlsx includes data on 204 employees at the (fictional) company Beta Technologies. For this problem, consider this data set as the population frame.**

a. Using the method in this section (not StatTools), generate a simple random sample of size 20 from this population.

b. Use StatTools to generate 10 simple random samples of size 20 from this population.

c. Calculate the population mean, median, and standard deviation of Annual Salary. Then calculate the sample mean, median, and standard deviation of Annual Salary for each of the samples in parts a and b. Comment briefly on how they compare to each other and the population measures.

**The file P07_02.xlsx contains data on the 1995 students who have gone through the MBA program at State University. You can consider this the population of State University’s MBA students.**

a. Find the mean and standard deviation for each of the numerical variables in this population. Also, find the following proportions: the proportion of students who are male, the proportion of students who are international (not from the USA), the proportion of students under 30 years of age, and the proportion of students with an engineering undergrad major.

b. Using the method in this section (not StatTools), generate a simple random sample of 100 students from this population, and find the mean and standard deviation of each numerical variable in the sample. Is there any way to know (without the information in part a) whether your summary measures for the sample are lower or higher than the (supposedly unknown) population summary measures?

c. c. Use StatTools to generate 10 simple random samples of size 100. For each, find the mean of School Debt and its deviation from the population mean in part a (negative if it is below the population mean, positive if it is above the population mean). What is the average of these 10 deviations? What would you expect it to be?

d. d. We want random samples to be representative of the population in terms of various demographics. For each of the samples in part c, find each of the proportions requested in part a. Do these samples appear to be representative of the population in terms of age, gender, nationality, and undergrad major? Why or why not? If they are not representative, is it because there is something wrong with the sampling procedure?

**A manufacturing company’s quality control personnel have recorded the proportion of defective items for each of 500 monthly shipments of one of the computer components that the company produces. The data are in the file P07_07.xlsx. The quality control department manager does not have sufficient time to review all of these data. Rather, she would like to examine the proportions of defective items for a sample of these shipments. For this problem, you can assume that the population is the data from the 500 shipments.**

a. Use Excel to generate a simple random sample of size 25 from the data.

b. Calculate a point estimate of the population mean from the sample selected in part a. What is the sampling error, that is, by how much does the sample mean differ from the population mean?

c. Calculate a good approximation for the standard error of the mean.

d. Repeat parts b and c after generating a simple random sample of size 50 from the population. Is this estimate bound to be more accurate than the one in part b? Is its standard error bound to be smaller than the one in part c?

**The manager of a local fast-food restaurant is interested in improving the service provided to customers who use the restaurant’s drive-up window. As a first step in this process, the manager asks his assistant to record the time it takes to serve a large number of customers at the final window in the facility’s drive-up system. The results are in the file P07_08.xlsx, which consists of nearly 1200 service times. For this problem, you can assume that the population is the data in this file**.

A. Use Excel to generate a simple random sample of size 30 from the data.

b. Calculate a point estimate of the population mean from the sample selected in part a. What is the sampling error, that is, by how much does the sample mean differ from the population mean?

c. Calculate a good approximation for the standard error of the mean.

d. If you wanted to halve the standard error from part c, what approximate sample size would you need? Why is this only approximate?

**Calculate the following probabilities using Excel. (If you have Excel 2010 or later, we suggest using its new functions.)**

a. P(t10 ≥ 1.75), where t10 has a t distribution with 10 degrees of freedom.

b. P(t100 ≥ 1.75), where t100 has a t distribution with 100 degrees of freedom. How do you explain the difference between this result and the one obtained in part a?

c. P(Z ≥ 1.75), where Z is a standard normal random variable. Compare this result to the results obtained in parts a and b. How do you explain the differences in these probabilities?

d. P(t20 ≤ −0.80), where t20 has a t distribution with 20 degrees of freedom.

e. P(t3 ≤ −0.80), where t3 has a t distribution with 3 degrees of freedom. How do you explain the difference between this result and the result obtained in part d?

**The file P08_05.xlsx contains salary data on all NFL players in each of the years 2002 to 2009. Because this file contains all players for each of these years, you can calculate the population mean for each year if population is defined as all NFL players that year. However, proceed as in the previous chapter to select a random sample of size 50 from the 2009 popula- tion. Based on this random sample, calculate a 95% confidence interval for the mean NFL total salary in 2009. Does it contain the population mean? Repeat this procedure several times until you find a random sample where the population mean is not included in the confidence interval.**

**The manager of a local fast-food restaurant is inter- ested in improving the service provided to customers who use the restaurant’s drive-up window. As a first step in this process, the manager asks an assistant to record the time (in seconds) it takes to serve a large number of customers at the final window in the facil- ity’s drive-up system. The file P08_07.xlsx contains a random sample of 200 service times during the busiest hour of the day.**

a. Identify the relevant population.

b. Construct and interpret a 95% confidence interval for the mean service time of all customers arriving during the busiest hour of the day at this fast-food operation.

c. If the manager wants to improve service, at least during the busiest time of day, does this confidence interval provide useful information? What useful information does it not provide?

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