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In a manufacturing process the assembly line speed (feet per minute) was thought to affectthe number of defective parts found during the inspection process. To test this theory, managers devised a situation in which the same batch of parts was inspected visually at avariety of line speeds. They collected the following data.a. Develop the estimated regression equation that relates line speed to the number ofdefective parts found.b. At a. 05 level of significance, determine whether line speed and number of defectiveparts found are related.c. Did the estimated regression provide a good fit to the data?d. Develop a 95$\%$ confidence interval to predict the mean number of defective parts fora line speed of 50 feet per minute.

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Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Piedmont College

Cairn University

Oregon State University

Idaho State University

Lectures

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06:59

The manufacture of a certa…

03:56

Three assembly lines are u…

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In 2017 , ABC News reports…

in order to find an estimated regression equation as given here, we need to come up with a piece of one and be subzero value. So in order to find our beasts of one value, let me do that. Over here we are going to take thes some of the product of the differences between each of the, um the given X and Y values and their respective means divided by sum of square differences between each X value and export. So with this formula, we need to first find X bar and y bar. So X bar is going to be equal to the mean off all of this. So we get a value of, um 35 and why bar is equal to 17. So this we will get a sum of each individual X value minus 35 times each individual. Why value minus 17 over the sum of each individual X value minus 35 squared until we get a be so one value of negative point one for eight. And now we need to find the This is sub zero. So be so. One is equal to negative 10.1 for eight and the sub zero is equal to y bar minus piece of one times export. So this is equal to 17 minus negative 0.1 for eight times 35 which is equal to 22.174 So over here we get in answer to part A of why hat equals 22.174 plus minus 0.148 x. So this is the answer to part A. Next, we are asked to find, um, at a 0.5 global significance determine whether these two variables are related. So in order to do that, we need to do a we need to find the P value associate with an f te statistic. And these are our hypotheses are no hypothesis is that there is no relationship, whereas our alternative hypothesis is that there isn't a relationship and protesting this at an outflow 0.5 So the first thing we're going to come up with is thesame of squares Um, future error. So, on a different page, the sum of squares due to error is equal to the sum of each individual. Why value minus, um, the predicted value at that point square. So um, we get this using the following data. Let me just put up. So we're going to take some. So, Hearst, this is the predicted hope. This is the predicted values. So that is what comes out of, um, why hat equals, um, 22.174 minus 0.148 x. So those are the values there, and we're going to take each individual. Why? Value and subtract its corresponding predicted value from it and square it and take the sum of all those differences. So that and these would be the difference is squared. And now we're going to take the sum of everything in this column. So the sum of squares due to error, you believe this sum of squares due to error would equal 8.872 Um, 8.869 8.869 Um, And now with this, we can come up with, um, the mean square of the air. So let's first actually fill out that part of the Innova table Are Yeah, another table. Um, sum of squares do. Error is 8.869 And now we need to come up with the mean square of the air and the mean square of the error is equal to the sum of squares of the air over and minus two. And we have 1234566 values in our data set. So, six months two is four. So 8.869 divided by four is equal to two point 217 So the main square of errors is 2.217 degrees of freedom is equal to four. And now we should come up with a sum of squares for the total. So the sum of squares of the total is equal to Ah, the the difference between, um so the sum of the difference between each individual. Why value? Sorry, why Value and, um, the associate id y bar or not associated just the Y bar squared. So with this, we are going to get, um, a value of so first we need to calculate that y bar right? And we did that in the beginning, and we got that Y bar equals 17. So what we would do is take the difference between each individual. Why? Value 21 minus 17 19 minus 17. 15 minus 17. Square each of those values and then add the monk. So I'll just do one example. This would be 2021 minus 17 squared plus 19 minus 17 squared until you get to the end of the data set and the sum of all of these would be equal to 34. So this is equal to 34. And now to find, um, he sum of squares for the regression, we would take the sum of squares of the total and subtract the sum of the squares of the error from it. So that would be 34 minus, uh, 8.869 and we would get a value of 25 0.13 But now we need to find the means square of the regression. So the mean square of the regression is equal to the sum of squares of the regression divided by the number of independent variables. And we have one independent very low, because we have one. Why so, um, emerging independent variables. So this is equal to 25.13 divided by one, which is 25.13 This is one this 25.13 Now we need to come up with an F statistic, which is the mean square of their aggression, divided by the mean square of the air, just 25.13 But by 2.217 and we get an F statistic of 11.3 repeating. So now that we have figured out our F table, let's come up with a P value associate with this. So the P value with an F of 11.3 first degrees of freedom of four and one, we would get a P value equal to, um, less than 0.1 And because we're comparing against an Alfa of 0.5 um, we can reject the no. So what does that mean? That means that we conclude that there exists a relationship between the number of shares selling and the expected price. And now we have to come up with a um, we have to figure out whether the estimated regression provided a good fit for the data. So you have a new page. So to find whether the data provides a good fit, we have to come up with an R squared value the coefficient of determination. So this is equal to the sum of squares of the regression divided by the sum of squares of the total. And we figured out the sum of the squares of the regression is equal to Ah, 25.13 divided by the sum of the squares of the total, which is equal to 34 and we get a value of 0.7391 So what does this mean? It means that approximately 73.91% of the variability and expected price can be explain by a linear relationship with the number of shares that are being sold. So because it's 73.9% that's a pretty good fit. Um, so that's a pretty good fit. And now we have Thio develop in 95% confidence interval. Is this party 95% confidence interval? So this means that our alphas equal to one minus 10.95 which is 0.5 and to come up with a confidence interval, we're going to take our, um, point estimate, uh, plus or minus our the tea of Alfa over two times the standard deviation. Um, so in order to come up with the standard deviation of why Hat Star So let's come up with the standard deviation of White Hat star first, this is equal to the trailer. Standard deviation times the square root of one over end. Plus the given X value minus the mean X over the sum of the square. Differences for the X values. Um, so yeah. Ah. Okay. So what, this we're going to take, um, we have our ex star equaling 50 I believe. Yeah, 50 feet per minute. So this is equal to 50. We know that much. We know that we have How many elements in our data set? Six. Now we need to find our It's far. We did that in the beginning. 35. This is equal to 35 over, and you do now is come up with ah, standard deviation. The S is equal to the square of the main square of the air is equal to the square root of means where the air of 25.13 necessary means we're there to point to 17 which is equal to one point for 893 So this is equal to one point for 893 on. Now we can come up with center deviation of our point estimate, which is equal to 1.4893 times the square root of one over six. Plus, um, sorry. Give me one second trying to figure out this value. Yeah, So it would be 50. Minus 30 squared over. Ah, See some of each individual X value minus X bar squared. So this would be 28 minus 35 square 20 minutes. 35 square, 40 minus 35 square 30 minutes. 35 record 61. 65 square, 40 minutes. Um, 35 square. So, um, so on and so forth until we get a value of 1150 and we would ultimately get a all you over here, this would equal 0.196 And we would get a, um, standard deviation of one point for a 93 times the square root of one divided by six plus 0.196 And we would get this word ultimately be 0.896 Okay. 0.896 Now we need to come up with a t te statistic. So a tea at Alfa over to so Alpha's 0.25 and our degrees of freedom is equal to the number of elements that we have, which is six minus two, which is for so a T TE statistic at 0.0 to 5, with four degrees of freedom is equal to two 0.776 And the last thing we need to do is come up with a Y hat at 50. So this is equal to 22.174 minus 0.148 times 50 and we get a value of 14.76 So are 95% conference in trouble, equal to 14.76 plus or minus 2.776 times there a 0.6313 and this is in the range. 13.7 two 16.513

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