Problem

The data from exercise 3 follow. $$ \begin{arra…

11:23

Problem 33 Easy Difficulty

The data from exercise 2 follow.
$$
\begin{array}{lllllll}{x_{i}} & {3} & {12} & {6} & {20} & {14}\end{array}
$$
$$
\begin{array}{l|lllll}{55} & {40} & {55} & {10} & {15}\end{array}
$$
$$
\begin{array}{l}{\text { a. Compute an estimate of the standard deviation of } \hat{y}^{* \prime} \text { when } x=8 \text { . }} \\ {\text { b. Develop a } 95 \% \text { confidence interval for the expected value of } y \text { when } x=8 \text { . }}\end{array}
$$
$$
\begin{array}{l}{\text { c. Estimate the standard deviation of an individual value of } y \text { when } x=8 \text { . }} \\ {\text { d. Develop a } 95 \% \text { prediction interval for } y \text { when } x=8 \text { . }}\end{array}
$$

Answer

See explanation

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Video Transcript

33 were given a small little data set and we're supposed to find a few things. So the first thing that we need to find here's the dataset. So here's the X. Values in the column and the Y. Values and the B. Column. And were given the X. Star to be eight. So we're just making a prediction whenever X equals eight and we'll need the X. Bar. So what I did there is I just said it's the average from A one A five and then I'll need each a value minus the X. Bar squared. That's what I did here. So basically what that is is that's that three minus 11 squared B 64. And then um 12 minus 11 squared, six minus 11 squared 20 minus 11 squared. And then 14 minus 11 squared. And then I added those up I just did the sum of G. two To G six all need those eventually with the formula. Okay so I need a few things go to this war document here, we have this is what I'm trying to find and I need the S. And that's this times the square root of one over and one is five. And then that X. Bar stuff, that's what I needed here. So really now all I need is the S. And there's a long way and there's a short way I'm gonna do the short way on this one. So if we go to data since we have the technology and we'll go to data analysis and regression and the Y value range will go from B. One to B. Five. And then the X. Value range will go from A 12 A five and then we'll click enter and this is what I need right here. So that's my standard air if you open this up. So that's the standard air of the estimates. That's 875595. So let's go and write that down now. So now that's the s That's 8.77595. And then we multiply that by the square to 1/5 plus X minus X. Bar squared. So that's eight Where the X. Star is the eight and then -11. Well 8 -11 Squared is nine. So that's nine Over that summation that I was looking at here. So that's on this sheet here. So that's the 180. So over 1 80. Okay and let me show that work there. So that's 8 -11 sq. There we go. Okay? So then you plug that in the calculator and you should get 4.387975. Alright, so that is the first answer. Alright? We're gonna use that to find the 95% confidence interval. Well we need a T. Critical value, so T. Star and so we're gonna use the calculator can use the table too. But I'm gonna use a calculator here and I'm gonna go to them inverse T function that um Gives us everything we need. So go to second distribution on T. four and go to inverse T. This will give you your critical value. The area is alpha over two. So we want a 95% confidence interval, semis alpha five or a cut that in half and do five. And the degrees of freedom is in minus two. So five minus two is three. And that's gonna give us the negative. But remember it's positive and negative with the Confidence interval. So positive, negative 3.18 Um 2 4. So let's write that down. So positive negative 3.1824. And then we just plug that into the equation here. Now we do need y hat and we're gonna use that regression line for y. Hat. So look over here, the regression line Is 68 -3 X. So those are my coefficients there. So that's 68 -3 times the x. star was eight And that's going to give me 44. So that's my point estimates of 44 plus or minus the T. Star or the tl fever too, Which is 3.1824 times that standard error of the estimate, which was um four point 387975. Okay, So what that is is it's 44 plus or minus 13 point 9643 when we multiply that. So that is your margin of air and then you can expand that out. So 44 minus this margin of error would be 30 357. and then 44 plus the 13 9 is 57 point 9643. So that's your 95% confidence interval involving those. Okay, so now let's look at the prediction is slightly different Formula but it's pretty much the same thing. We have all the material actually. So the standard error of the prediction is the S which was that eight 75595 that we got from Excel and then times one plus one over N which is 1/5. And then plus that X star minus X. Bar. That's that eight minus 11 since the X bar is 11. And then divided by that 1 80. That summation that we did. All right. So whenever you do that, you should get the prediction Air as 97894. After you type that in. And we're going to use that for the 95% prediction. So we still have the point estimate of 40 for the critical value is still going to be 3.18 2 4. But this time the Prediction error is 97894. So that's going to be 44 plus or minus 31 point 1538. And then if you take 44 minus that 31, you get 12 .8462 And 44 plus the 31. You get 75 point 1538. Okay, so there's your 95 prediction interval.

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