In Exercises 21–24, a control chart is shown. Each chart has h...

Question

In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.

A gear has been designed to have a diameter of 3 inches. The standard deviation of the process is 0.2 inch.

A gear has been designed to have a diameter of 3 inches. The standard deviation of the process is 0.2 inch.

Control chart showing gear diameters over 10 observations with mean and control limits marked.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A chemical process is designed to produce a solution with a pH of 7.00. The standard deviation of the process is 0.10. Use the control chart below to determine whether the given process is in control or out of control. Awesome. So it appears for this particular problem we're asked to take our control chart that is provided to us by the prom itself, and we're asked to determine whether or not the given process is in control or out of control. So it's either going to be A in control or B out of control. So looking at our control chart, we can see that our Y axis, the vertical axis, represents the pH. And the X axis, the horizontal axis, represents the observation number. And our curve is represented with a solid black line with our different data points indicating our different observation numbers with their corresponding pH values. So, with that in mind, let us note that the prom itself tells us that the target means, the center line is equal to 7.00. And Let us call the mean because right once again this is the target mean and let's call the mean mu. So the mean mu, the target mean the center line is 7.00, and the standard deviation, which we'll call sigma, we're told that sigma by the pro itself is equal to 0.10. And once again, this is our standard deviation. And moving right along here. So the control limits will be as follows. So the upper control limit, which we'll call UCL, which once again the upper control limit, will be equal to 7.00. Plus, 3 multiplied by 0.10, which will mean that UCL, once again, the upper control limit will be equal to 7.30 when we perform that calculation. And the lower control limit, which we'll call LCL, will be equal to 7.00 minus 3 multiplied by 0.10, which when we perform that calculation, you will find that it's equal to 6.70. Awesome. So moving right along here. Our next step is we need to determine the two standard deviation lines. So we need to take 7.00. Plus or minus 2 multiplied by 0.10, which will be equal to square brackets of 6.80.7.20. Fantastic. So at this point, as we should recall a note, based on the information that's provided to us, there are 3 conditions used to determine whether a given process is in control or out of control. So, Firstly, Are there any points beyond 3 multiplied by sigma? These are, these would be values outside of 7.30 or 6.70. Our second condition would be, are there 9 consecutive points on one side of the mean, 7.00, and finally, our third condition, are Are there at least 2 out of 3 consecutive points beyond 2 multiplied by sigma from the mean, which would be outside of 6.80 to 7.20? These boundaries are shown as dashed lines on our graph. So when we analyze our graphs, so when we look at it again, looking at our dash dashed lines which are represented in red, we need to determine, are there any points beyond 3 multiplied by sigma that are outside of our 7.30 or 6.70 points? And when we do make this observation, we determine that yes, the observation.8. It is above 7.30, so the process is out of control. So therefore, without a doubt, in conclusion, the process is out of control due to rule one being triggered. So that will mean that our final answer without a doubt will be multiple choice answer letter B out of control, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.