Statistics

1. Case: Rare Events Part 1 in the textbook.

Study the case carefully and then answer one of the first four Questions for Thought at the end of the case study.

2. Case: Rare Events Part 2 in the textbook.

Using the same case answer either question 5 or 6 from the Questions for Thought at the end of the case study.

3. Select one of the following:

a. Review your chosen thesis and check if the sample size, validation of the sample, and the inferences are correct. Support your observation by citing examples.

Or

b. Shopping Downtown – Problem 38 in Chapter 15.

Consider each situation described below. Identify the population and the sample, explain what the parameter p or or m represents, and tell whether the methods of this chapter can be used to create a confidence interval. If so, find the interval.

(a) The service department at a new car dealer checks for small dents in cars brought in for scheduled maintenance. It finds that 22 of 87 cars have a dent that can be removed easily. The service department wants to estimate the percentage of all cars with these easily repaired dents.

(b) A survey of customers at a supermarket asks whether they found shopping at this market more pleasing than at a nearby store. Of the 2,500 forms distributed to customers, 325 were filled in and 250 of these said that the experience was more pleasing.

(c) A poll asks visitors to a website for the number of hours spent Web surfing daily. The poll gets 223 responses one day. The average response is three hours per day with s = 1.5.

(d) A sample of 1,000 customers given loans during the past two years contains 2 who have defaulted.

4. Insulator – Problem 35 in Chapter 14.

One stage in the manufacture of semiconductor chips applies an insulator on the chips. This process must coat the chip evenly to the desired thickness of 250 microns or the chip will not be able to run at the desired speed. If the coating is too thin, the chip will leak voltage and not function reliably. If the coating is too thick, the chip will overheat. The process has been designed so that 95% of the chips have a coating thickness between 247 and 253 microns. Twelve chips were measured daily for 40 days, a total of 480 chips.

(a) Do the data meet the sample size condition if we look at samples taken each day?

(b) Group the data by days and generate X-bar and S-charts with control limits at {3 SE. Is the process under control?

(c) Describe the nature of the problem found in the control charts. Is the problem an isolated incident (which might be just a chance event), or does there appear to be a more systematic failure?

5. 4M Monitoring an Email System – Problem 37 in Chapter 14.

A firm monitors the use of its email system. A sudden change in activity might indicate a virus spreading in the system, and a lull in activity might indicate problems on the network. When the system and office are operating normally, about 16.5 messages move through the system on average every minute, with a standard deviation near 8.

The data for this exercise count the number of messages sent every minute, with 60 values for each hour and eight hours of data for four days (1,920 rows). The data cover the period from 9 A.M. to 5 P.M. The number of users on the system is reasonably consistent during this time period.

Motivation

(a) Explain why the firm needs to allow for variation in the underlying volume. Why not simply send engineers in search of the problem whenever email use exceeds a rate of, say, 1,000 messages?

(b) Explain why it is important to monitor both the mean and the variance of volume of email on this system.

Method

(c) Because the computer support team is well staffed, there is minimal cost (aggravation aside) in having someone check for a problem. On the other hand, failing to identify a problem could be serious because it would allow the problem to grow in magnitude. What value do you recommend for a, the chance of a Type I error?

(d) To form a control chart, accumulate the counts into blocks of 15 minutes rather than use the raw counts. What are the advantages and disadvantages of computing averages and SDs over a 15-minute period compared to using the data for 1-minute intervals?

Mechanics

(e) Build the X-bar and S-charts for these data with a = 0.0027 (i.e., using control limits at {3 SE). Do these charts indicate that the process is out of control?

(f) What is the probability that the control charts in part (e) signal a problem even if the system remains under control over these four days?

(g) Repeat part (e), but with the control limits set according to your choice of a. (Hint: If you used a = 0.0027, think harder about part (c).) Do you reach a different conclusion?

Message

(h) Interpret the result from your control charts (using your choice of a) using nontechnical language. Does a value outside the control limits guarantee that there’s a problem?