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In exercise 18 , the data on grade point average $x$ and monthly salary $y$ provided the esti-mated regression equation $\hat{y}=1790.5+581.1 x .$$$\begin{array}{l}{\text { a. Develop a } 95 \% \text { confidence interval for the mean starting salary for all students with a }} \\ {3.0 \text { GPA. }}\end{array}$$$$\begin{array}{l}{\text { b. Develop a 95 } \% \text { prediction interval for the starting salary for Joe Heller, a student with }} \\ {\text { a GPA of 3.0. }}\end{array}$$

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Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Temple University

Oregon State University

University of St. Thomas

Idaho State University

Lectures

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12:53

In exercise 18 the data on…

05:59

The following data are the…

01:21

Age, GPA, and Income A res…

05:43

The data from exercise 2 f…

02:12

In exercise $1,$ the follo…

04:42

Based on the regression ou…

10:31

Refer to exercise $21,$ wh…

01:13

The accompanying data show…

01:04

STATISTICS For Exercises 3…

01:23

For Exercises $13,15,$ and…

02:26

Average Value of a Functio…

01:22

The given values represent…

06:38

Attitudes In the study of …

04:11

Ideal proportions The stud…

00:45

Use the chart at the right…

00:41

For the following exercise…

01:26

08:04

Use the data in WAGE2 to e…

00:48

The average, or mean, $A,$…

problem number 35 arrest to find a 95% confidence interval. Uh For someone that has a three uh G. P. A. Of three finding what their their income level is. And here's the regression model here 17 90.5 plus 5 81.1 x. Okay so here's our data set. Here's the GPA. And then the associated uh salary and the X. Star. That's the data value of concern. That's the 3.0 GPA. So three. And then what I did here so I pre did some things but I found the X. Bar. The mean of the sample for the GPA. So the mean GPA is 3.2. So I just got that by saying equals average A 12 A six. And then over here I did the X. I minus X. Bar squared because I'll need this it's part of my standard Air. And so I did a one minus the 3.2 squared and then you know A A two minus 3.2 squared and so on and so forth. And then I added those up. So I did some of G. Two to G. Seven. Okay so I have most of the stuff in there. The other thing that I need, I'll go to data analysis and then regression and my Y. Input ranges B one to B. Six. And then for X it's a one to a six. And I press OK And this standard air is what I'm looking at. So that 1:45.8896 is what I need. That's my s so here's what I'm trying to find. I need to find this in order to get my 95% confidence interval. So that that s was 1 45 point 8896. And we multiply that by the square root One over the sample size. Where there were six data values there and then plus X star which was at three minus X bar, which is 32 squared divided by the summation. Well, actually already have this. So it's the summation of X I minus X bar. And if you look back at that excel Sheet there, it's .74. So we're dividing that by .74. Okay, so when we do that, you should get about 68 0.54. Okay, so now we're ready for our confidence in the rule. So the confidence center will I don't know if I wrote that down or not? No, I didn't. So the confidence interval is why hat star plus or minus the T alpha over two. I called T Star times the S Y. Hat star. We already found this piece. Quickly. Find this piece here in a second and then this is a calculated thing. So let's find the critical value first because that's the easiest. We need to go to inverse norm. I'm not in first or I'm sorry, inverse t. And the area is gonna be point oh 25 because it's a 95% confidence interval. It's the 5% divided by 2/2. The degrees of freedom here is gonna be four and then we paste this is going to give us the negative but it's the same thing. So it's the positive and negative. 2.776. Okay so that's the T. Star. Alright so 2.77 6 4. And then we multiply that by these six 68.54. So that's going to give us our margin of air. Now we need the point estimate. So why hat is where we just plug in three for this X here. So why hat star is 1795 Plus 5 81.1 Times three. And so we should get if you plug that in correctly you should get 35 33.8. So that's our point estimate for someone that makes it 3.0 GPA. So we have 35, 33 appointed. Thank. And whenever you simplify that you should get 35 33.8 Plus or minus this multiplied together is 1 90 30 and You take 35 33.8 -1 90.3 You get 33 43.5 And then you add the margin of error and you should get 37 24.1. Okay so that's our 95% confidence interval and then we do a prediction interval basically the exact same thing except now we have a one in there. So the s spread okay, is still The 145 .8896. But this time we multiplied by the square root of one Plus 1/6 Plus that 3 -3.2 squared Over the 74. So the s spread is 161 188. And whenever we multiply that by the critical value, So we still have the same 35, plus or minus the Critical value is still the 2.77 64. I mean multiply that by the 161.188 and that gives you 35 33 8 plus or minus 4 47.5. And then whenever you expand that out, take 35 33 8 -4475. You get 30 86.3 And then you take the point estimate plus the margin of error there and you should get 30 9 81 .3. So there's our 95% prediction interval given this data

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