Question

The number of serious nonconformities in twenty-five samples of 5 automobiles are

recorder in the Excel File Question 3. Can you conclude that the process is in
statistical control? What center line and control limits would you recommend for
controlling future production?
4. The tensile strength of a certain metal component is normally distributed with a
mean of 10,000 kilograms per square centimeter and a standard deviation of 100
kilograms per square centimeter. If specifications require that all components have
tensile strength between 9800 and 10,200 kilograms, what proportion of pieces
would we expect to meet specifications?

Answer 1

To determine if the process is in statistical control, we can use control charts. I would recommend using the p-chart to monitor the number of nonconformities in the samples. The center line and control limits can be calculated based on the average number of nonconformities and the variability in the data. For future production, the center line can be used as a target and the control limits can be used to monitor the process. To calculate the proportion of pieces that meet specifications, you can use the z-score formula and a standard normal distribution table or calculator.

Step-by-step explanation:

To determine if the process is in statistical control, we can use control charts. Control charts are used to monitor processes over time and determine if they are within statistical control. A commonly used control chart for count data is the p-chart. In this case, the number of serious nonconformities is the count data. We can calculate the center line and control limits for the p-chart using the average number of nonconformities and the variability in the data.

To determine the center line, calculate the average number of nonconformities over the 25 samples. To calculate the control limits, we need to determine the upper control limit (UCL) and lower control limit (LCL). These can be calculated using the formula:

UCL = Average number of nonconformities + 3 * √((Average number of nonconformities * (1 - Average number of nonconformities)) / (25 * 5))

LCL = Average number of nonconformities - 3 * √((Average number of nonconformities * (1 - Average number of nonconformities)) / (25 * 5))

By comparing the number of nonconformities in each sample to the control limits, we can determine if the process is in statistical control.

For future production, I would recommend using the center line as a target for the average number of nonconformities. The control limits can be used to monitor the process and identify any deviations from the target.

As for the proportion of pieces that will meet specifications, we can use the z-score formula to calculate the proportion within a specified range. The z-score is calculated as:

z = (x - μ) / σ

where x is the value we want to find the proportion for, μ is the mean, and σ is the standard deviation. In this case, we want to find the proportion of pieces with a tensile strength between 9800 and 10200 kilograms. We can calculate the z-scores for both values and use them to determine the proportion using a standard normal distribution table or a calculator.