Quality Associates, Inc., a consulting firm, advises its clients about sampling and statistical procedures that can be used to control their manufacturing processes. In one particular application, a client gave Quality Associates a sample of 800 observations taken during a time in which that client's process was operating satisfactorily. The sample standard deviation for these data was 0.21; hence, with so much data, the population standard deviation was assumed to be 0.21. Quality Associates then suggested that random samples of size 30 be taken periodically to monitor the process on an ongoing basis. By analyzing the new samples, the client could quickly learn whether the process was operating satisfactorily. When the process was not operating satisfactorily, corrective action could be taken to eliminate the problem. The design specification indicated the mean for the process should be 12. The hypothesis test suggested by Quality Associates follows.

H₀: = 12

H_a: 12

Corrective action will be taken any time H₀ is rejected.

A) Prepare a managerial report to establish process control limits for a process with a mean value of 12 at a 0.01 level of significance.

B)The samples in the following data set were collected at hourly intervals during the first day of operation of the new statistical process control procedure.

Sample 1	Sample 2	Sample 3	Sample 4
12.01	11.90	11.56	11.61
12.01	11.35	11.63	11.68
12.04	11.74	11.53	11.58
12.17	11.94	11.76	11.81
12.10	12.13	11.91	11.96
12.06	11.71	11.65	11.70
12.04	11.60	11.81	11.86
11.63	11.84	12.04	12.09
12.38	12.15	11.95	12.00
11.64	11.90	11.93	11.98
12.10	12.11	12.14	12.19
11.89	11.60	12.10	12.15
12.21	12.20	11.94	11.99
11.87	11.55	12.22	12.27
12.02	11.94	12.33	12.38
12.34	12.00	11.94	11.99
12.08	12.05	11.86	11.91
11.76	11.75	11.77	11.82
12.19	11.81	12.17	12.22
11.78	12.11	11.78	11.83
12.29	11.59	12.01	12.06
12.26	11.94	12.05	12.10
12.28	11.95	11.99	12.04
12.46	12.21	12.31	12.36
12.02	11.74	12.19	12.24
12.16	11.95	11.98	12.03
11.93	11.94	12.18	12.23
11.96	11.88	11.86	11.91
12.22	11.87	12.31	12.36
12.24	11.92	12.16	12.21

C)Compute the standard deviation for each of the four samples.

sample 1s=

sample 2s=

sample 3s=

sample 4s=

The assumption of 0.21 for the population standard deviation appears reasonable for which of the following samples. (Select all that apply.)

a)Sample 1

b)Sample 2

c)Sample 3

d)Sample 4

e)The assumption of 0.21 is not reasonable for any of the samples.

D)Conduct a hypothesis test for each sample at the = 0.01 level of significance.

Use the sample standard deviations in calculating the test statistic. (You may need to use the appropriate appendix table or technology to answer this question. Round your test statistics to two decimal places and your p-values to four decimal places.)

Sample 1

Find the test statistic and p-value.

test statistic=

p-value=

State your conclusion. (Choose one)

a)Reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

b)Reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

c)Do not reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

d)Do not reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

Sample 2

Find the test statistic and p-value.

test statistic=

p-value=

State your conclusion. (choose one)

a)Reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

b)Reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

c)Do not reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

d)Do not reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

Sample 3

Find the test statistic and p-value.

test statistic=

p-value=

State your conclusion. (choose one)

a)Reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

b)Reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

c)Do not reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

d)Do not reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

Sample 4

Find the test statistic and p-value.

test statistic=

p-value=

State your conclusion.(choose one)

a)Reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

b)Reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

c)Do not reject H₀. There is sufficient evidence to conclude that the mean is significantly different from 12.

d)Do not reject H₀. There is insufficient evidence to conclude that the mean is significantly different from 12.

Based on the hypothesis tests, which sample(s) need corrective action? (Select all that apply.)

a)Sample 1

b)Sample 2

c)Sample 3

d)Sample 4

e)No samples need corrective action.

D)Compute a 99% confidence interval for the population mean = 12 such that, as long as a new sample mean is within those limits, the process will be considered to be operating satisfactorily. If x exceeds the upper limit or if x is below the lower limit, corrective action will be taken. These limits are referred to as upper and lower control limits for quality control purposes. to

Compute the sample mean for each of the four samples.

sample 1 x=

sample 2 x=

sample 3 x=

sample 4 x=

Based on the sample means and control limits, which sample(s) need corrective action? (Select all that apply.)

a)Sample 1

b)Sample 2

c)Sample 3

d)Sample 4

e)No samples need corrective action.

E)Discuss the implications of changing the level of significance to a larger value. What mistake or error could increase if the level of significance is increased?

Increasing the level of significance will cause the null hypothesis to be rejected [SELECT: less OR more] often. Although this may mean quicker corrective action when the process is [SELECT: in OR out of] control, it also means there will be a higher error probability of [SELECT: continuing OR stopping] the process when the process [SELECT: is OR is not] operating satisfactorily. This would be an increase in the probability of a making a type [SELECT: I OR II] error.