Case Study Homework Problem: Quality Associates, Inc., a consulting firm, advise
ID: 3217691 • Letter: C
Question
Case Study Homework Problem: Quality Associates, Inc., a consulting firm, advises its clients about sampling and statistical procedures that can be used to control their manufacturing processes. In an 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 the sample size of 30 should be taken randomly and periodically to monitor the quality process on an ongoing basis. By analyzing the new 4 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 Quality Associate two-tailed hypotheses are as follows: H_(o: µ= 12 ) H_(a: µ 12) If the null hypothesis is rejected, corrective action needs to be taken. Please download the Excel file (Quality), which has the data of four samples. Answer the following questions: Conduct a hypothesis test for each of the four samples at the .01 level of significance and determine what action, if any should be taken. Provide the test statistic and p-value for each of the four tests in the four samples. Compute the standard deviation for each of the four samples. Does the assumption of .21 for the population standard deviation appear reasonable, or does it approximate near this value number? Why or why not? Discuss the implications of changing the level of significance to a larger value. What type of error could increase if the level of significance is increased? Explain.
Must have number for each of the following:
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 11.90 11.97 12.14 12.11 11.64 11.71 11.72 12.07 11.80 11.87 11.61 12.05 12.03 12.10 11.85 11.64 11.94 12.01 12.16 12.39 11.92 11.99 11.91 11.65 12.13 12.20 12.12 12.11 12.09 12.16 11.61 11.90 11.93 12.00 12.21 12.22 12.21 12.28 11.56 11.88 12.32 12.39 11.95 12.03 11.93 12.00 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 12.20 11.77 11.84 12.12 11.79 12.00 12.07 11.60 12.30 12.04 12.11 11.95 12.27 11.98 12.05 11.96 12.29 12.30 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 12.30 12.37 11.88 12.23 12.15 12.22 11.93 12.25Explanation / Answer
Solution:
For the given scenario, we have to check the hypothesis or claim whether the population mean is 12 or not. We are given a population standard deviation, so we have to use the one sample z test for the population mean. The null and alternative hypothesis for this test is given as below:
Null Hypothesis: H0: µ = 12
Alternative Hypothesis: Ha: µ 12 (This is a two tailed test.)
Alternative Hypothesis: Ha: µ < 12 (This is a lower tailed or left tailed test.)
Alternative Hypothesis: Ha: µ > 12 (This is a upper tailed or right tailed test.)
We consider the level of significance for this test as = 0.05 or 5%.
For the given sample, we have
Process mean = µ = 12
Sample size = n = 30
Degrees of freedom = n – 1 = 30 – 1 = 29
Sample mean = Xbar = 11.96
Sample standard deviation = S = 0.220356
Population standard deviation = = 0.21
The test statistic formula is given as below:
Z = (Xbar - µ) / [ / sqrt(n)]
Z = (11.96 – 12) / [0.21/sqrt(30)]
Z = -1.0433
Standard error = [ / sqrt(n)]
Standard error = 0.21/sqrt(30) = 0.0383
Lower critical value = -1.96
Upper critical value = 1.96
Alpha = 0.05
Z/2 = 1.96
P-value = 0.2968 (two tailed)
P-value = 0.8516 (Upper tailed)
P-value = 0.1484 (Lower tailed)
All p-values are greater than the given level of significance or alpha value, so we do not reject the null hypothesis at 5% level of significance.