Control Charts for p. In Exercises 5–12, use the given process...

Question

Control Charts for p. In Exercises 5–12, use the given process data to construct a control chart for p. In each case, use the three out-of-control criteria listed near the beginning of this section and determine whether the process is within statistical control. If it is not, identify which of the three out-of-control criteria apply.

Euro Coins Consider a process of minting coins with a value of one euro. Listed below are the numbers of defective coins in successive batches of 10,000 coins randomly selected on consecutive days of production.

32 21 25 19 35 34 27 30 26 33

Accepted Answer

Hello, everyone. Let's take a look at this question together. Construct a control chart, or P using the provided process data and answer the following question. Does the control chart indicate that the production process is statistically controlled, if not identify the out of control criteria that apply? A factory manufactures bolts that must adhere to strict quality standards. The number of defective bolts found in successive batches of 5000 bolts randomly sampled over consecutive days is as follows 1822, 1627, 1925, 30, 21, 23, and 28. Is it answer choice A? No, the process is out of control because at least one point falls outside the control limits. Answer choice B, no. The process is out of control because there is a clear increasing trend in the number of defective bolts. Answer choice C. No, the process is out of control because there is a long run of consecutive points on one side of the center line, or answer choice D, yes, the process is statistically controlled because all points lie within the control limits and there are no trends. So in order to solve this question, we have to recall. What we have learned about constructing a control chart to determine if the control chart would indicate that the production process is statistically controlled, and if it is not, we have to identify the out of control criteria that apply. And the first step in constructing a control chart for P to assess whether the process is statistically controlled, we must first compute the proportion of defective bolts for each sample. which the proportion defective PI for each sample can be calculated using the formula where PI is equal to the number of defective bolts in a sample divided by the sample size, which based on the information that we are given, we know the sample size is 5000. So we calculate the proportion defective for each sample, which gives us the following results. And then using this data, we can Compute the average proportion defective, which is P, and this can be calculated using the formula P is equal to the sum of all PI values divided by 10, which using the data that we calculated, we can determine that the average proportion defective value is 0.00458, and then using these values, we can determine the standard deviation of P using the formula for the standard deviation. Of P, which provides the value of 0.00955, and then we can calculate the upper control limit and the lower control limit where the formula for the upper control limit is calculated by T bar plus 3 times the standard deviation of P, which plugging in the values that we already calculated, we get 0.007445 as the value for the upper control. Limit and the formula for the lower control limit is P minus 3 times the standard deviation of P, which plugging in the values gives us 0.00172 as the value for the lower control limit. So now we can plot the control chart using the values for the center line, which is the value for P bar of 0.00458. We know the upper control limit is 0. 00744, and the lower control limit is 0.00172, as well as the data points that we can plot on our control chart, which are the proportion defective or the PI values, and we are given the following control chart, which looking at our control chart to analyze if we are out of control, which we use our three standard out of control criteria, starting. With any point outside of the control limits, we can identify that all values are between 0.00172 and 0.00744. Therefore, no points are outside of the limits. Then we are checking to see if there is a run of consecutive points on one side of the center line, which we can identify the values do not stay above or below P bar for an extended run, and lastly, We are checking for a trend of increasing or decreasing values over several points, which from our control chart, we can identify there is no clear upward or downward trend, and therefore we do not meet the criteria for any of the standard out of control criteria. Therefore, our control chart does in fact indicate that the process is statistically controlled, and there are no immediate signs of deterioration. Which makes answer choice D the correct answer. As answer choice D says yes, the process is statistically controlled because all points lie within the control limits, and there are no trends, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Control Charts

Out-of-Control Criteria

Defective Rate (p)

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