Lab

7. 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?

8. 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?

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

a. P(−2.00 ≤ t10 ≤ 1.00), where t10 has a t distribution with 10 degrees of freedom

b. P(−2.00 ≤ t100 ≤ 1.00), 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(−2.00 ≤ Z ≤ 1.00), 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. Find the 68th percentile of the t distribution with 20 degrees of freedom.

e. Find the 68th percentile of the t distribution with 3 degrees of freedom. How do you explain the difference between this result and the result obtained in part d?

22. The file P08_06.xlsx contains data on repetitive task times for each of two workers. John has been doing this task for months, whereas Fred has just started. Each time listed is the time (in seconds) to perform a routine task on an assembly line. The times shown are in chronological order.

a. Calculate a 95% confidence interval for the standard deviation of times for John. Do the same for Fred. What do these indicate?

b. Given that these times are listed chronologically, how useful are the confidence intervals in part a? Specifically, is there any evidence that the variation in times is changing over time for either of the two workers?