Test Statistic and P-Value

In the last application, we have received a sample of 800 observations taken during a time in which the process was operating satisfactorily. The sample standard deviation for this data was 0.21; the population standard deviation was assumed to be 0.21. Then we suggested that random samples of size 30 be taken periodically to monitor the process on an ongoing basis. The design specification indicated the mean for the process should be 12. The hypothesis test suggestion is as follows:

H0: ?=12

Ha: ? ?12

Test statistic and p-value

We need to compare the p-value to the level of significance to see whether the null hypothesis should be rejected or not. According to the sample data, we do not reject Ho in the first 3 samples because the p-value is greater than 0.01. But the p-value of sample 4 is 0.0003, which is smaller than 0.01 (level of significance). Hence, we reject sample 4.

With a level of significance of ?=0.01, the statistical evidence doesn't support the conclusion that the process was not operating satisfactorily in the first 3 samples. Hence, no corrective action should be taken to eliminate the problem. However, the data of sample 4 provide sufficient evidence to conclude that the process was not operating satisfactorily. We are 99% confident that the mean is differs from 12. The evidence against the mean being 12 is very strong in sample 4. And we can conclude that the alternative hypothesis, Ha: M ? 12 is true. In this case, statistical evidence supports the conclusion that the process was not operating satisfactorily. Therefore, the corrective action is warranted for sample 4.

Standard deviation

We wish to make a statement about the population standard deviation, whether it appears reasonable or not, according to the samples used. Since the sample standard deviations for all four samples are in the 0.19 to 0.21 range. It appears that the process population standard deviation assumption of 0.21 is good.

Control Limits

With level of significance at ?=0.01, z0.005 is equal to 2.576. Using the standard error of the mean of 0.04, the result of upper and lower control limits are as follows:

Upper Control Limit: