Statistics for Business Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

Chapter 20

Statistical Methods for Quality Control - all with Video Answers

Educators

Chapter Questions

Problem 1

A process that is in control has a mean of $\mu=12.5$ and a standard deviation of $\sigma=.8$
a. Construct the $\bar{x}$ control chart for this process if samples of size 4 are to be used.
b. Repeat part (a) for samples of size 8 and 16
c. What happens to the limits of the control chart as the sample size is increased? Discuss why this is reasonable.

Trinity Steen

Trinity Steen

Numerade Educator

Problem 2

Twenty-five samples, each of size $5,$ were selected from a process that was in control. The sum of all the data collected was 677.5 pounds.
a. What is an estimate of the process mean (in terms of pounds per unit) when the process is in control?
b. Develop the $\bar{x}$ control chart for this process if samples of size 5 will be used. Assume that the process standard deviation is .5 when the process is in control, and that the mean of the process is the estimate developed in part (a).

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 3

Twenty-five samples of 100 items each were inspected when a process was considered to be operating satisfactorily. In the 25 samples, a total of 135 items were found to be defective.
a. What is an estimate of the proportion defective when the process is in control?
b. What is the standard error of the proportion if samples of size 100 will be used for statistical process control?
c. Compute the upper and lower control limits for the control chart.

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 4

A process sampled 20 times with a sample of size 8 resulted in $\bar{X}=28.5$ and $\bar{R}=1.6$ Compute the upper and lower control limits for the $\bar{x}$ and $R$ charts for this process.

Trinity Steen

Numerade Educator

Problem 5

5. Temperature is used to measure the output of a production process. When the process is in control, the mean of the process is $\mu=128.5$ and the standard deviation is $\sigma=.4$
a. Construct the $\bar{x}$ chart for this process if samples of size 6 are to be used.
b. Is the process in control for a sample providing the following data?
$$128.8 \quad 128.2 \quad 129.1 \quad 128.7 \quad 128.4 \quad 129.2$$
c. Is the process in control for a sample providing the following data?
$$129.3 \quad 128.7 \quad 128.6 \quad 129.2 \quad 129.5 \quad 129.0$$

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 6

A quality control process monitors the weight per carton of laundry detergent. Control limits are set at $\mathrm{UCL}=20.12$ ounces and $\mathrm{LCL}=19.90$ ounces. Samples of size 5 are used for the sampling and inspection process. What are the process mean and process standard deviation for the manufacturing operation?

Trinity Steen

Numerade Educator

Problem 7

The Goodman Tire and Rubber Company periodically tests its tires for tread wear under simulated road conditions. To study and control the manufacturing process, 20 samples, each containing three radial tires, were chosen from different shifts over several days of operation, with the following results. Assuming that these data were collected when the manufacturing process was believed to be operating in control, develop the $R$ and $\bar{x}$ charts.

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 8

Over several weeks of normal, or in-control, operation, 20 samples of 150 packages each of synthetic-gut tennis strings were tested for breaking strength. A total of 141 packages of the 3000 tested failed to conform to the manufacturer's specifications.
a. What is an estimate of the process proportion defective when the system is in control?
b. Compute the upper and lower control limits for a $p$ chart.
c. With the results of part (b), what conclusion should be made about the process if tests on a new sample of 150 packages find 12 defective? Do there appear to be assignable causes in this situation?
d. Compute the upper and lower control limits for an $n p$ chart.
e. Answer part (c) using the results of part (d).
f. Which control chart would be preferred in this situation? Explain.

Trinity Steen

Numerade Educator

Problem 9

An automotive industry supplier produces pistons for several models of automobiles. Twenty samples, each consisting of 200 pistons, were selected when the process was known to be operating correctly. The numbers of defective pistons found in the samples follow.
\[
\begin{array}{rrrrrrrr}
8 & 10 & 6 & 4 & 5 & 7 & 8 & 12 & 8 & 15 \\
14 & 10 & 10 & 7 & 5 & 8 & 6 & 10 & 4 & 8
\end{array}
\]
a. What is an estimate of the proportion defective for the piston manufacturing process when it is in control?
b. Construct the $p$ chart for the manufacturing process, assuming each sample has 200 pistons.
c. With the results of part (b), what conclusion should be made if a sample of 200 has 20 defective pistons?
d. Compute the upper and lower control limits for an $n p$ chart.
e. Answer part (c) using the results of part (d).

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 10

For an acceptance sampling plan with $n=25$ and $c=0,$ find the probability of accepting a lot that has a defect rate of $2 \% .$ What is the probability of accepting the lot if the defect rate is $6 \% ?$

Trinity Steen

Numerade Educator

Problem 11

Consider an acceptance sampling plan with $n=20$ and $c=0 .$ Compute the producer's risk for each of the following cases.
a. The lot has a defect rate of $2 \%$
b. The lot has a defect rate of $6 \%$

Trinity Steen

Numerade Educator

Problem 12

Consider an acceptance sampling plan with $n=20$ and $c=0 .$ Compute the producer's risk for each of the following cases.
a. The lot has a defect rate of $2 \%$
b. The lot has a defect rate of $6 \%$

Trinity Steen

Numerade Educator

Problem 13

Refer to the KALI problem presented in this section. The quality control manager requested a producer's risk of .10 when $p_{0}$ was .03 and a consumer's risk of .20 when $p_{1}$ was .15 . Consider the acceptance sampling plan based on a sample size of 20 and an acceptance number of $1 .$ Answer the following questions.
a. What is the producer's risk for the $n=20, c=1$ sampling plan?
b. What is the consumer's risk for the $n=20, c=1$ sampling plan?
c. Does the $n=20, c=1$ sampling plan satisfy the risks requested by the quality control manager? Discuss.

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 14

To inspect incoming shipments of raw materials, a manufacturer is considering samples of sizes $10,15,$ and $20 .$ Use the binomial probabilities from Table 5 of Appendix $B$ to select a sampling plan that provides a producer's risk of $\alpha=.03$ when $p_{0}$ is .05 and a consumer's risk of $\beta=.12$ when $p_{1}$ is .30

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 15

A domestic manufacturer of watches purchases quartz crystals from a Swiss firm. The crystals are shipped in lots of $1000 .$ The acceptance sampling procedure uses 20 randomly selected crystals.
a. Construct operating characteristic curves for acceptance numbers of $0,1,$ and 2 .
b. If $p_{0}$ is .01 and $p_{1}=.08,$ what are the producer's and consumer's risks for each $\operatorname{sam}$ pling plan in part (a)?

Rowan Ahmed

Numerade Educator

Problem 16

Samples of size 5 provided the following 20 sample means for a production process that is believed to be in control.
\[
\begin{array}{lll}
95.72 & 95.24 & 95.18 \\
95.44 & 95.46 & 95.32 \\
95.40 & 95.44 & 95.08 \\
95.50 & 95.80 & 95.22 \\
95.56 & 95.22 & 95.04 \\
95.72 & 94.82 & 95.46 \\
95.60 & 95.78 &
\end{array}
\]
a. Based on these data, what is an estimate of the mean when the process is in control?
b. Assume that the process standard deviation is $\sigma=.50 .$ Develop the $\bar{x}$ control chart for this production process. Assume that the mean of the process is the estimate developed in part (a).
c. Do any of the 20 sample means indicate that the process was out of control?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 17

Product filling weights are normally distributed with a mean of 350 grams and a standard deviation of 15 grams.
a. Develop the control limits for the $\bar{x}$ chart for samples of size $10,20,$ and 30
b. What happens to the control limits as the sample size is increased?
c. What happens when a Type I error is made?
d. What happens when a Type II error is made?
e. What is the probability of a Type I error for samples of size $10,20,$ and $30 ?$
f. What is the advantage of increasing the sample size for control chart purposes? What error probability is reduced as the sample size is increased?

Beth Stone

Numerade Educator

Problem 18

Twenty-five samples of size 5 resulted in $\bar{x}=5.42$ and $\bar{R}=2.0 .$ Compute control limits for the $\bar{x}$ and $R$ charts, and estimate the standard deviation of the process.

Rashmi Sinha

Numerade Educator

Problem 19

The following are quality control data for a manufacturing process at Kensport Chemical Company. The data show the temperature in degrees centigrade at five points in time during a manufacturing cycle. The company is interested in using control charts to monitor the temperature of its manufacturing process. Construct the $\bar{x}$ chart and $R$ chart. What conclusions can be made about the quality of the process?

Dominador Tan

Numerade Educator

Problem 20

The following were collected for the Master Blend Coffee production process. The data show the filling weights based on samples of 3 -pound cans of coffee. Use these data to construct the $\bar{x}$ and $R$ charts. What conclusions can be made about the quality of the production process?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 21

Consider the following situations. Comment on whether the situation might cause concern about the quality of the process.
a. $A p$ chart has $L C L=0$ and $U C L=.068 .$ When the process is in control, the proportion defective is $.033 .$ Plot the following seven sample results: .035, .062, .055, .049 $.058, .066,$ and $.055 .$ Discuss.
b. $\quad$ An $\bar{x}$ chart has $L C L=22.2$ and $U C L=24.5 .$ The mean is $\mu=23.35$ when the process is in control. Plot the following seven sample results: 22.4,22.6,22.65,23.2 $23.4,23.85,$ and $24.1 .$ Discuss.

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 22

Managers of 1200 different retail outlets make twice-a-month restocking orders from a central warehouse. Past experience shows that $4 \%$ of the orders result in one or more errors such as wrong item shipped, wrong quantity shipped, and item requested but not shipped. Random samples of 200 orders are selected monthly and checked for accuracy.
a. Construct a control chart for this situation.
b. Six months of data show the following numbers of orders with one or more errors: 10 $15,6,13,8,$ and $17 .$ Plot the data on the control chart. What does your plot indicate about the order process?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 23

An $n=10, c=2$ acceptance sampling plan is being considered; assume that $p_{0}=.05$ and $p_{1}=.20$
a. Compute both producer's and consumer's risk for this acceptance sampling plan.
b. Would the producer, the consumer, or both be unhappy with the proposed sampling plan?
c. What change in the sampling plan, if any, would you recommend?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 24

An acceptance sampling plan with $n=15$ and $c=1$ has been designed with a producer's risk of .075
a. Was the value of $p_{0} .01, .02, .03, .04,$ or $.05 ?$ What does this value mean?
b. What is the consumer's risk associated with this plan if $p_{1}$ is $.25 ?$

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 25

A manufacturer produces lots of a canned food product. Let $p$ denote the proportion of the lots that do not meet the product quality specifications. An $n=25, c=0$ acceptance sampling plan will be used.
a. Compute points on the operating characteristic curve when $p=.01, .03, . .10,$ and $.20 .$
b. Plot the operating characteristic curve.
c. What is the probability that the acceptance sampling plan will reject a lot containing .01 defective?

Dominador Tan

Numerade Educator