Question 1197979: 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 following 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 data set Quality.
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.
Compute the standard deviation for each of the four samples. Does the assumption of .21 for the population standard deviation appear reasonable?
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.
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?
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! **1. Hypothesis Testing for Each Sample**
* **Hypotheses:**
* **H0:** μ = 12 (The process mean is 12)
* **H1:** μ ≠ 12 (The process mean is not 12)
* **Test Statistic:**
* Since the population standard deviation (σ) is known, we use the z-test.
* z = (x̄ - μ) / (σ / √n)
* where:
* x̄ is the sample mean
* μ is the population mean (12)
* σ is the population standard deviation (0.21)
* n is the sample size (30)
* **Calculate z-statistic and p-value for each sample.**
* **Sample 1:**
* Calculate x̄ for Sample 1.
* Calculate z = (x̄ - 12) / (0.21 / √30)
* Use a z-table or statistical software to find the p-value associated with the calculated z-statistic.
* **Repeat for Sample 2, Sample 3, and Sample 4.**
* **Decision Rule:**
* If p-value ≤ 0.01, reject H0.
* If p-value > 0.01, fail to reject H0.
* **Action:**
* If H0 is rejected, the process is likely out of control. Take corrective action.
* If H0 is not rejected, the process is likely in control.
**2. Sample Standard Deviations**
* Calculate the standard deviation for each of the four samples.
* Compare these sample standard deviations to the population standard deviation (0.21).
* If the sample standard deviations are consistently close to 0.21, the assumption of a population standard deviation of 0.21 appears reasonable.
* If the sample standard deviations are significantly different from 0.21, the assumption may not be valid.
**3. Control Limits**
* **Calculate the standard error of the mean:**
* Standard Error (SE) = σ / √n = 0.21 / √30
* **Calculate the control limits:**
* Upper Control Limit (UCL) = μ + (z-value * SE)
* Use the z-value corresponding to the desired level of significance (e.g., for a 99% confidence level, z-value ≈ 2.576)
* Lower Control Limit (LCL) = μ - (z-value * SE)
* **Monitoring:**
* Continuously monitor the process by collecting new samples.
* Calculate the sample mean for each new sample.
* If the sample mean falls outside the control limits (above UCL or below LCL), investigate and take corrective action to bring the process back in control.
**4. Implications of Changing the Level of Significance**
* **Increasing the level of significance (e.g., from 0.01 to 0.05) increases the probability of rejecting the null hypothesis when it is actually true.**
* This is known as a **Type I error** (false positive).
* In this context, a Type I error would mean incorrectly concluding that the process is out of control when it is actually operating satisfactorily.
* This could lead to unnecessary investigations, adjustments, and potential disruptions to the production process.
**Note:**
* This analysis requires the actual data from the "Quality" data set to perform the calculations and draw conclusions.
* Statistical software like Excel, R, or statistical packages can be used to efficiently perform the calculations and generate the necessary reports.
This framework provides a foundation for implementing a statistical process control system. By continuously monitoring the process and taking corrective action when necessary, you can improve product quality, reduce waste, and increase efficiency.
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