QUESTION 5. (Data file included: Yes; In Sheet: Quality): 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 while 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 analysing the new samples, the client could 5 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 that the mean for the process should be 12. Corrective actions should be taken is the mean of the process does not equal to 12. 1. First set up the hypothesis to be tested. Clearly show the null and the alternative hypothesis. 2. Conduct a hypothesis test for each sample at the 0.05 level of significance and determine what action, if any, should be taken. Provide the test statistic and p value for each test. 3. Compute the standard deviation for each of the four samples. Does the conjecture of 0.21 for the population standard deviation appear reasonable?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
QUESTION 5.
(Data file included: Yes; In Sheet: Quality):
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 while 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 analysing the new samples, the client could 5 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 that the mean for the process should be 12. Corrective actions should be taken is the mean of the process does not equal to 12.
1. First set up the hypothesis to be tested. Clearly show the null and the alternative hypothesis.
2. Conduct a hypothesis test for each sample at the 0.05 level of significance and determine what action, if any, should be taken. Provide the test statistic and p value for each test.
3. Compute the standard deviation for each of the four samples. Does the conjecture of 0.21 for the population standard deviation appear reasonable?
4. 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. Use α = 0.05. 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.
5. 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?
6. Quality Associates now believes that if the mean of the process from the sample exceeds 12, then they won’t as alarmed if it was to go below 12. Conduct a hypothesis test under this scenario for each sample at the 0.05 level of significance and determine what corrective action, if any, should be taken in case the sample results in mean values less than the expected. Provide the test statistic and p value for each test.
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