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. The hypothesis test suggested by Quality Associates follows.

Corrective action will be taken any time  is rejected.

The samples on the following page were collected at hourly intervals during the first day of operation of the new statistical process control procedure. These data are available in the file 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 the standard deviation for each of the four samples. Does the assumption of .21 for the population standard deviation appear reasonable?
3. Compute limits for the sample mean  around  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.
4. 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?

Discussion

Initial post prompt: Your Managerial Report serves as your initial post to the discussion forum. After responding to the requirements posed by the Managerial Report, also provide an example in your career in which you believe one of the lessons learned from the Case has been/could be applicable. Alternatively, if you don’t have/foresee direct experience relevant to your current position, what type of scenario can you anticipate occurring where you can utilize one of the lessons learned from examining this case?

Response post prompt: Consider Managerial Reports posted by two of your peers. One or both of your responses may be to Managerial Reports for a case problem different from your own. Think critically and ask open-ended questions. If you agree, consider their position and expand upon their ideas. Provide an additional perspective. If you disagree, provide your reasoning. Always be professional and courteous in your responses.

Post by classmate 1

Hi everyone,
This week, I am bringing the analysis of Case Problem1: Quality Associates, Inc. The table provides four samples of 30 hourly intervals and the design specification indicated the mean for the process should be equal to 12. Corrective action will be taken any time the null hypothesis is rejected. The calculations are the following:

Sample 1Sample 2Sample 3Sample 4

Sample Mean 11.96
Standard Deviation 0.2204

z-score -1.027

P-value 0.3044

Sample Mean 12.03
Standard Deviation 0.2204

z-score 0.713

P-value 0.4759

Sample Mean 11.89
Standard Deviation 0.2072

z-score -2.935

P-value 0.0033

Sample Mean 12.08
Standard Deviation 0.2061

z-score 2.161

P-value 0.0307

Considering the 0.01 significance level, Sample 1 presents a p-value higher value and we should not reject the null hypothesis. The same case happens with samples 2 and 4 and corrective actions are not needed. However, sample 3 presents a p-value lower than the significance level so we must reject the null hypothesis and request corrective actions.

Based on the four samples, the standard deviation for each was close to .21. Two samples were above and two samples were below, so .21 is reasonable.
Lower Limit = 11.9012

Upper Limit = 12.0988

By using a superior value for the significance of a higher value, we are making the confidence interval smaller. This increases the probability of making a type one error. In summary, only sample three indicated that corrective action was needed. This was discovered by comparing the p-value to the upper tail area at the 99% confidence level and was further encouraged by the confidence interval that was found.

References

Anderson, D. R. (2020). Modern business statistics with Microsoft Excel (7th ed.). Cengage.

Post by classmate 2

Q1: For this weeks managerial report I decided to do the report on Quality Associates. From the information given, we can determine that the level of significance is .01, the assumed standard deviation is .21 and the hypothesized value of the date is 12. Using this information, we can find the Test Statistic and the P-Value. From that data we determine that we fail to reject the null hypotheses in all of the samples because the P-Value is greater than the .01 L.O.S. Sample 1: .8479 Sample 2: .2381 Sample 3: .9983 Sample 4: .0153.

Q2: The standard deviations for all four samples is as follows:

Sample 1Sample 2Sample 3Sample 4Sample Size30303030Sample Mean11.9612.0311.8912.08Sample St.Dev0.220.220.210.21

The assumption of .21 for the population standard deviation is reasonable for this set of data.

Q3: The lower and upper limit for each sample can be seen below:

Sample 1Sample 2Sample 3Sample 4Lower Limit11.865111.935111.799711.9920Upper Limit12.052312.122311.978312.1707

From this information, we can determine if each sample is operating satisfactorily or if corrective actions need to be addressed. If x exceeds the upper limit or if x is below the lower limit corrective action is needed. With a m=12 we find that samples 1,2 and 4 are operating satisfactorily but Sample 3, the mean exceeds the upper limit of that sample meaning corrective action is needed.

Q4: If the level of significance is increased then the rejection becomes more likely. Meaning more corrective actions would need to be taken would have to happen. a type 1 error would also increase with the level of significance increasing.

In the sales world, I do not see how hypothesis test would be a very beneficial tool. While I sell products and services, my goal isn’t necessarily to figure out when errors will take place with products. That information typically falls to people who are more technical than I am and have a much deeper understanding of the products we sell.

Anderson, Sweeney, Williams. Modern Business Statistics with Microsoft Excel. 7th edition, 2021. Cengage. ISBN-10:
0357131509 | ISBN-13: 9780357131503. You might wish to consider Cengage Unlimited.