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Problem 1 Estimate the process capability using x-bar and R charts for the power supply voltage data below. If the specifications are 350*5, calculate the Cp,Cpk, C...

Question

Problem 1 Estimate the process capability using x-bar and R charts for the power supply voltage data below. If the specifications are 350*5, calculate the Cp,Cpk, Cpkn and the confidence interval of Cp and Cpk Interpret these capability ratios. Here Xi = (observed voltage on u niti 350) X 10Sample Number X1XzXX4102 3 4 5 61015 8 10 6 1 10 7 13 10 10 10 9 5 10 8 16 13 12 " 13 ; 13 10 1Q 10 8 1Q ; 6 9 14 HI 159 ; 8 9 9 10 15 # 8 6 16 3 I5 16 17 18 : 19 20 13

Problem 1 Estimate the process capability using x-bar and R charts for the power supply voltage data below. If the specifications are 350*5, calculate the Cp,Cpk, Cpkn and the confidence interval of Cp and Cpk Interpret these capability ratios. Here Xi = (observed voltage on u niti 350) X 10 Sample Number X1 Xz X X4 10 2 3 4 5 6 10 15 8 10 6 1 10 7 13 10 10 10 9 5 10 8 16 13 12 " 13 ; 13 10 1Q 10 8 1Q ; 6 9 14 HI 15 9 ; 8 9 9 10 15 # 8 6 16 3 I5 16 17 18 : 19 20 13



Answers

The lengths of power failures, in minutes, are recorded in the following table. $\begin{array}{rrrrrrrrr}22 & 18 & 135 & 83 & 55 & 28 & 70 & 66 & 74 \\ 40 & 98 & 87 & 50 & 96 & 118 & 15 & 90 & 78 \\ 121 & 120 & 13 & 89 & 103 & 24 & 132 & 115 & 21 \\ 158 & 74 & 78 & 69 & 22 & 21 & 28 & 83 & 98 \\ 102 & 124 & 112 & 120 & 121 & 43 & 37 & 93 & 95\end{array}$ (a) Find the sample mean and sample median of the power-failure times. (b) Find the sample standard deviation of the powerfailure times.

The following is a solution for # three. And we're asked to find a 980% confidence interval given this this data here. So we're gonna use the formula that I have up in red here, it's p one minus p two hats. In this case it's 20.255 -0.193 plus or minus Now the Z score here. If you don't have it memorized, that's fine. In fact, I can show you how to get it real quick because this one isn't as typical. So 1995-99 are usually the most typical ones. But if we go to distribution here and inverse norm And we type in .8 And then the center will be center is always for confidence intervals and then we paste, we get positive negative 1.282. So that's what we're gonna use for our z value. So positive, negative 1.282 times the square root of 0255 Times 1 -255. Which is 0.745 Divided by that sample size of 300 Plus 019, three Times 1- that. Which is 0807. And that's divided by the other sample size of 400 and you should get 0.062 plus or -0.041. So you can leave it like that. Or you can actually expand it out Where you subtract and you add you should get 0.021. Oops, Okay. All the way up to 0.103. Okay, so either answer is acceptable. Now we find a 95% confidence interval Given this following data. So we started out the same way now 95% confidence that we should know that Z value that's 196. But let's go and start this .1 47 which is the P one hat minus 10.131 Plus or -1.96 which is the critical value for 95 times the square root of .147 Times 1 -147 which is 853. And then divide that by the sample size of 3500 And then plus .131 Times 1 -131, which is 869. And divide that by the sample size of 37 50. So whenever you do that you should get .016 plus or minus .0159. So you can leave it like this certainly. Or you can expand it out groups Where you get essentially zero, also point 0001 All the way up to 0.0319. Okay, so those are two representations of the 95% confidence interval. Given this data

The following is a solution to number four and we're asked to find the 99.5% confidence interval for the difference of two population means given the following data here, and we're gonna use a two sample T interval. The reason why we have to use a T intervals because the sample size isn't greater than 30. So if at least one of these is not Uh at least one of these sample sizes less than 30, then we've got to use the tea interval instead of the z interval. So I'm gonna use a tt for here. If you got a stat, There are over two tests and it's gonna be option zero. So that's all the way down here where it says to samp t ent press center and just make sure data, I'm sorry, stats is highlighted and you can just punch in your data there. The confidence level is 00.995 stands for 99 99.5%. And then under pooled, you're just gonna want to keep that is Yes. The reason being, is that the two standard deviations are the same. Whenever that happens, then we have to have a pulled experiment And then we calculate in this top band here is our confidence intervals, so 10 4-9 and 14571. Okay, so we're 99.5% confident that the true population mean difference is between 10 4-9 and 14 .571. Okay. Mhm. And part B. It's very similar. We just want to be a little more confidence in 99.9% confident for the difference to population means again we're gonna use the tea interval because this 25 that's not greater than 30. So if we go to again stat and then tests and then the zero option. Now I actually have changed these because it's different data obviously. So 215 is the X one, bar seven is the standard deviation and 25 is that sample size? And then 1 85 is that I mean 12, is that standard deviation? And then 35 was the second sample size and we want to be .999. So 99.9% confident pull this still yes. And then we calculate and we get between 20.713 and 39.287. Okay, So let's go and write that down. 27 13 all the way up to 39 0.28 7. 99.9% confident that the true difference of the population means is between these two numbers. Mhm

The following is a solution number one, and we're asked to find the 95% confidence interval for the difference of two population means given this data. Now, these sample sizes are less than 30, which means we need to use the tea interval, the two sample t interval if at least one of these is less than 30 and they both are in this case. So I'm gonna use the society for here. And if you go to stat and then tests and it's the show you here is the two sample TNT's. So if you just press zero or scroll up or down to two, samp a T ent click on that and then just make sure stats is highlighted here, you can see your data. So X one bar S for the standard deviation sample size and so on and then the sea levels 10.95 not. You get to this thing called pooled and for this section it's always gonna be yes, because it actually says in the in the very top directions, assume that the standard deviations are the same population. Standard deviations are the same. So whenever that's the case, then you're going to have a pooled, So you just click Yes there and then we calculate and that gives us 16.16-21.8. Force, that's our confidence intervals. All right up here. So 16 .161 to 21 point 839, So that's Are 95% confidence interval from difference of two population means. And then similar here, we find the 99% confidence and we're using this data so again we're gonna use the T interval because these sample sizes are so small and we don't know what the population standard deviation is. So go back to stat tests and then like I said, it's the zero option and we're gonna type in our stuff here. So X one bar is 25. This standard deviation was one, The sample size for the first sample was only six, and then the X- two bar was 17. And then that standard deviation was three and then that sample size was 12 and we're asked to find the 99%ile. So .99. Again we're back at yes and calculate and that gives us 4.2767-11.723. Okay, so let's write that down. Four point 2767. groups All the way up to 11 7-3. So that's The 99% confidence interval for the difference of two population means. Using the TI 84 calculator

The following is a solution for number two and we're asked to find a 90% confidence interval for the difference of two population means given this data now, since the two sample sizes are less than 30 or at least one of them is less than 30 we have to use the two sample t interval instead of the z interval. So I'm gonna use a graphing calculator here. If we go to stat and then test and we are all the way down to the zero option where it says to samp T and that stands for two sample T interval and then we click enter and then just make sure stats is highlighted and we punch in our data here, so X one, bar, s one in one in the next two bar S two and into so that's all the data that was there and the confidence levels nine for 90%. Then you get to this pulled and just make sure it's yes, because it says that that assume that the two standard deviations are the same. So whenever that's the case, then you can go ahead and pull them and then we calculate, and we get this 11.371 to 16.6 to nine, It's just gonna write that down. So 11 .371 To 16, 6- nine. Yes, that's the 90% confidence interval. And then similarly we're gonna do a 99% confidence interval, still using the t interval since they're both Or at least one of them is less than 30 with the sample size. And so if we go to stat and then tests and then just go to the zero option And we need to change this data here. So 68 was Rx one, bar eight was Rs one and then 14 was our first sample size and then uh 43 was our second sample mean three was our standard deviation and 20 was that sample size? And we want to be 99% confidence. And again, we're gonna be pooled. So we say calculate and we get 19.657 to 30.343. Okay, so 19.6 57 2, 4 3. So we're 99% confident that the difference of the two means Given the sample data is between these two numbers.


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