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The data from exercise I follow.$$\frac{x_{i}}{y_{i}} \left| \begin{array}{ccccc}{1} & {2} & {3} & {4} & {5} \\ \hline y_{i} & {3} & {7} & {5} & {11} & {14}\end{array}\right.$$$$\begin{array}{l}{\text { a. Use equation }(12.23) \text { to compute an estimate the standard deviation of } \hat{y}^{*} \text { when } x=4 \text { . }} \\ {\text { b. Use expression }(12.24) \text { to develop a } 95 \% \text { confidence interval for the expected value }} \\ {\text { of } y \text { when } x=4 .}\end{array}$$$$\begin{array}{l}{\text { c. Use equation }(12.26) \text { to estimate the standard deviation of an individual value of } y} \\ {\text { when } x=4 .} \\ {\text { d. Use expression }(12.27) \text { to develop a } 95 \% \text { prediction interval for } y \text { when } x=4 \text { . }}\end{array}$$

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Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Piedmont College

Oregon State University

Idaho State University

Boston College

Lectures

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05:43

The data from exercise 2 f…

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In Exercises 1–4, find x a…

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$$\begin{array}{c|c}x …

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For the following exercise…

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$$\text { Is }(-1,-2) \tex…

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In Exercises 7-10, find $x…

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In Exercises $51-54,$ grap…

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02:25

In Exercises 1–4, make a c…

$$\text { In Exercises } 1…

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In Exercises $1-16,$ calcu…

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$\begin{array}{l}{\text { …

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Use the following table to…

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In the following exercises…

01:18

So we're gonna the following information. I went ahead and computed the mean for the exes, which is one plus two plus three plus four plus five divided by five. So we get an export of three. And because we know this is from exercise number one and exercise number one, we figured out the estimated regression equation. Is that why had equal 0.2 plus 2.6 X. So I found out each of the individual Why has, um, over here And with this we can go forward in the problem. First thing we have to compute is tthe e estimate of the standard deviation of why hats star when X is equal to four. So here's the formula for the estimate of the standard deviation four y hat. So the first thing we need to compute is this s over here. We have to figure out what that is in formula for s is equal to the square root of the means. Squares of the air. That means square of the error is equal to the sum of squares of the air divided by and minus two. And is the number of elements in our set Um, and some of squares of the air is equal to the, um, difference between or the sum of the difference some of the difference between each individual, Why value that we have and the expected y value based on our regression equation squared. So let's find this sum of squares of the error first, and we get you a new page. And the sum of squares of the air is equal to three minus 2.8 squared plus seven minus 5.4 squared. Plus the difference between all the individual by why values and their corresponding. Why hats? Um and eventually we get some of the squares of the air equaling 12.4. And now we need to come up with the mean square of the air, which is equal to the sum of squares of the air divided by the number of elements we have minus two. We have five elements here. Five elements and five minus two is equal to three. So 12.4 divided by three is equal to four point 13 repeating. And now we can come up with the standard deviation which is equal to the square root of 4.13 repeating equal to 2.331 And now we have one part of our equation. So this is equal to 2.331 And we also know that our N is equal to five. And we know that Ah, this extra represents the value that were given the independent value. And we are estimating wife hat when X is equal to four. So this is equal to four and our ex bar, we already calculated to be three. Our exit by is each of our individual X value. So we're given that and our expert is equal to three. So using this information, we can come up with, um, the estimated standard deviation for my hat when X is equal to four. So this is equal to 2.331 times the square root of one over our end, which is five plus. The difference between our are given value, which is four minus our mean, which is three squared, divided by the sum of each of our individual X values and three squared. So this over here, if we expand it is equal to, um one minus three. Squared two minus three, squared three minus three, squared four minus three. Squared five minus three squared. Um, all some together. So one minus three squared plus two minus three squared plus three bodies s three squared. Plus all the way to five minus three square. And we're going to take some of this. And with that, we get a total of 10 down here. So, um, erase all of this. I don't want to get that. Ah, break. No. Okay, great. So and this is 10 down here. And now with this, we have to just do Ah, the basic math. And we get that the sum of or this standard deviation of why I had to star when X is equal to four is equal to 1.113575 And now, using another equation, we have to come up with, uh, the confidence interval a 95% conference interval. The expected value of why when x is equal to four. So in order to find our confidence interval Ah, 95% confidence interval. So this means that our Alfa is equal to one minus our confidence level of 0.95 So we have an Alfa 0.5 Um, And now using this, we should come up with, um, our white hat star, which is just, uh, the estimate of why hat given our x value and were given that our X value is equal to four. So this is equal to 10.6, and now we have to compute. So let me just give you the formula for the, um, conference interval first. So that is equal to why Hat Star, plus or minus the T's test statistic on Alfa over four Alfa over two times the standard deviation of Wie hat star The estimated standard deviation of what head start. We figured this already. This is equal to 1.113575 Ah, we've just figured out why. Hat Star, That should be a star. Um, this is equal to 10.6. Now we just need to figure out a T test statistic for Alfa over two eso our tea at 0.0 to 5 and we have three degrees of freedom because our end is five. So are and it's five and our degrees of freedom is n minus too. So we have three degrees of freedom. So given that we have three degrees of freedom at an Alfa over two of 0.25 we get a T value of 3.182 So this confidence interval is equal to this conference is terrible is equal to, um 10.6 plus or minus 3.182 times 1.113575 which ultimately leads to Ah, a interval of 7.566 to 14.1434 So what does this mean? It means that we're 95% confident that the true value of why hat or the expected value of why when X is equal to four is between these two numbers. And now we have to come up with the standard deviation of an individual wow value of why when x is equal to four. And that is pretty similar to what we had before. So the Senate deviation of an individual value is equal to our regular standard deviation times the square root of one over end. All right. One plus one over end plus the difference between are given X and our ex bar squared, divided by the square root of our individual X values and our ex bar differences squared. So we already figured out what Oh, this information is and what our s is so we can just fill in values Put things into 2.331 kind of squared of one plus won over five plus four minus three squared over the square root of this stuff and we could get out out to be 10 and we get a value of 2.3181 Okay. And now we have to come up with a 95% prediction in trouble for why, when x is equal to four. So the prediction interval is very similar to, ah, the other conference interval. We have at least formula wise. It's equal to the white hat, um, star plus or minus the tea at Alfa over too times the standard deviation of the individual. So our Alfa is equal to one minus 10.95 one, minus nine point point 95 which is equal to 0.50 point 05 divided by two is 20.25 We already figured out this, um, this value events and we already figured out our white hat. It's our when X is equal to four, so that is equal to 10.6, 10.6 plus or minus. Uh, our ah t value of 3.182 times stand deviation of the individual, which is 2.3181 And then we get a confidence interval of 3.2 to 38 to 17 0.9762 So what does this mean? It means that we're 95% confident, confident that or we predict with 95% confidence that the expected value of why when X is equal to four is between 3.2 to 38 to 17.9.

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