Question

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 .21; hence, with so much data, the population standard deviation was assumed to be .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. Corrective action will be taken any time H₀ is rejected.

The samples were collected at hourly intervals during the first day of operation of the new statistical process control procedure. These data are available in the Excel file named Quality.

Managerial Report

1. Conduct a hypothesis test for each sample at the .01 level of significance and determine what action, if any, should be taken. Provide the test statistic and p-value for each test.
2. Compute limits for the sample mean  around μ = 12 such that, as long as a new sample mean is within those limits, the process will be considered to be operating satisfactorily. If  exceeds the upper limit or if  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.
3. 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?
   Sample 1 Sample 2 Sample 3 Sample 4 11.62 12.02 11.55 11.91 11.36 12.02 11.62 11.69 11.52 11.59 12.05 11.75 11.82 12.18 11.75 11.95 11.90 12.14 11.97 12.11 11.64 11.72 12.07 11.71 11.6 11.87 12.05 11.80 12.03 11.85 11.64 12.10 12.16 12.39 11.94 12.01 11.99 11.9 11.65 11.92 12.20 12.13 12.12 12.11 12.16 12.09 11.61 11.90 12.00 11.93 12.21 12.22 12.21 12.28 11.56 11.88 12.32 12.39 12.03 11.95 12.00 12.01 11.93 12.35 12.06 12.09 11.85 11.92 11.83 11.76 11.77 11.76 12.16 12.23 11.82 12.20 11.77 11.84 11.79 12.12 12.00 12.07 11.60 12.30 12.04 12.27 12.11 11.95 11.98 12.05 12.29 11.96 12.30 12.37 12.22 12.47 12.18 12.25 11.75 12.03 12.04 11.96 12.17 11.97 12.17 12.24 11.95 11.94 11.85 11.92 11.89 11.97 12.30 12.23 12.37 11.88 12.15 12.22 11.93 12.25
   

Answer 1

	Sample 1	Sample 2	Sample 3	Sample 4
	11.55	11.62	11.91	12.02
	11.62	11.69	11.36	12.02
	11.52	11.59	11.75	12.05
	11.75	11.82	11.95	12.18
	11.9	11.97	12.14	12.11
	11.64	11.71	11.72	12.07
	11.8	11.87	11.61	12.05
	12.03	12.1	11.85	11.64
	11.94	12.01	12.16	12.39
	11.92	11.99	11.91	11.65
	12.13	12.2	12.12	12.11
	12.09	12.16	11.61	11.9
	11.93	12	12.21	12.22
	12.21	12.28	11.56	11.88
	12.32	12.39	11.95	12.03
	11.93	12	12.01	12.35
	11.85	11.92	12.06	12.09
	11.76	11.83	11.76	11.77
	12.16	12.23	11.82	12.2
	11.77	11.84	12.12	11.79
	12	12.07	11.6	12.3
	12.04	12.11	11.95	12.27
	11.98	12.05	11.96	12.29
	12.3	12.37	12.22	12.47
	12.18	12.25	11.75	12.03
	11.97	12.04	11.96	12.17
	12.17	12.24	11.95	11.94
	11.85	11.92	11.89	11.97
	12.3	12.37	11.88	12.23
	12.15	12.22	11.93	12.25
Mean	11.95867	12.02867	11.889	12.08133

Hypothesis testing:

The provided sample mean is X = 11.95866667 and the known population standard deviation is O = 0.21, and the sample size is n = 30 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho:= 12 Ha: 12 This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.01, and the critical value for a two- tailed test is ze = 2.58. The rejection region for this two-tailed test is R {z : |z| > 2.58} (3) Test Statistics The z-statistic is computed as follows: 11.95866667 -12 -1.078 0.21/30

(4) Decision about the null hypothesis Since it is observed that z= 1.078 rejected ze = 2.58, it is then concluded that the null hypothesis is not Using the P-value approach: The p-value is p = 0.281, and since p = 0.281 > 0.01, it is concluded that the null hypothesis is not rejected (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean u is different than 12, at the 0.01 significance level. Confidence Interval The 99% confidence interval is 11.86 < u < 12.057

The first sample is within the control limits.

The confidence or control limits are given by

The provided sample mean is X = 12 and the population standard deviation is o 0.21. The size of the sample is n = 30 and the required confidence level is 99% Based on the provided information, the critical z-value for a 0.01 is ze 2.576 O = The 99% confidence for the population mean u is computed using the following expression CI =(X X+ n Therefore, based on the information provided, the 99 % confidence for the population mean u is 2.576 x 0.21 12+ 2.576 x 0.21 12 CI = 30 V30 = (12 - 0.099, 12+0.099) (11.901, 12.099)

Only sample 3 has mean lower than the lower limit of the confidence interval.

Thus we will reject the null hypothesis for sample 3. The procedure is the same as done above. We have done the hypothesis testing for sample 3 below.

(3) Test Statistics The z-statistic is computed as follows: X 11.889 12 -2.895 0.21//30 (4) Decision about the null hypothesis Since it is observed that |z| = 2.895 > ze 2.58, it is then concluded that the null hypothesis is rejected Using the P-value approach: The p-value is p 0.0038, and since p = 0.0038 < 0.01, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean u is different than 12, at the 0.01 significance level.

Rest of the samples are within the control limits.