Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

In exercise $8,$ data on $x=$ temperature rating $\left(\mathrm{F}^{\circ}\right)$ and $y=$ price (S) for 11 sleeping bagsmanufactured by Bergans of Norway provided the estimated regression equation $\hat{y}=$$359.2668-5.2772 x .$ For these data $s=37.9372 .$$$\begin{array}{l}{\text { a. Develop a point estimate of the mean price for all sleeping bags with a temperature }} \\ {\text { rating of } 30 .} \\ {\text { b. Develop a } 95 \% \text { confidence interval for the mean price for all sleeping bags with a tem- }} \\ {\text { perature rating of } 30 \text { . }}\end{array}$$$$\begin{array}{l}{\text { c. Suppose that Bergans developed a new model with a temperature rating of } 30 . \text { Develop }} \\ {\text { a } 95 \% \text { prediction interval for the price of this new model. }} \\ {\text { d. Discuss the differences in your answers to parts (b) and (c). }}\end{array}$$

See explanation

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Missouri State University

University of North Carolina at Chapel Hill

University of St. Thomas

Idaho State University

Lectures

0:00

05:43

The data from exercise 2 f…

05:28

In exercise $1,$ the follo…

10:31

Refer to exercise $21,$ wh…

05:59

The following data are the…

02:12

07:48

Refer to the data presente…

01:32

A mathematical model that …

03:50

(a) Use the method describ…

02:23

For Exercises 103–107, ass…

02:42

Find the regression equati…

06:33

The following problem is r…

01:30

For the following exercise…

01:11

Driving speed and fuel con…

06:14

Let's try the re-expr…

01:29

13:32

For the data in Exercises …

01:15

Fitting a Line to Data In …

01:31

01:13

06:43

The following data were us…

problem number 36 we're going to make a confidence interval. And the prediction interval. This is the sleeping bags example. They give us the regression line of three 59.2668 -5.2772 x. and then the standard area here of 37.9372. So first off let's just find a point estimate where X equals 30. And that's just a simple plugging in 30 for X. And you get So I'll just go and show the work here. But you plug in 34 the X. And you should get the point estimate. So why hat Equals about 201. So this is the like the rating or something based on the temperature rating. I don't know. I can't remember exactly what it is. Okay, so now we find the confidence interval in that formula is why I had Why star had if you want to it's just that to one plus or minus the T. Alpha over two star and then times the s why star. Hat So the alpha over to the T off over two. That's the critical value. That's easy enough to find. You can use a table or a calculator. In fact let's just go and do that right now and use a calculator. So I'm gonna use inverse T. Okay, so in verse T. And the area if we want 95% of the Alphas .05 but you cut that in half. So it's .25. And then the degrees of freedom. There were 11 data values here. So degrees of freedom will be nine And that gives us two point that gives us the negative but it's gonna be positive negative 2-6 to to So that's the Critical values at two 2 6- two. Right now we have this thing. So let's find what the Sequels. So this equals the s which was that 37.9372 And then times the square root of one. Over the end. The end was the sample size. There's 11 of them. And then we have this thing. So over here in my excel, I kind of already did this. So here are the X values. This is really the only ones I care about. I think these are the temperatures And then I went ahead and find X. Bar. So I found the average of those. So average of a 1-8 11. And then what I did was I took each of these individual data values. So the average is 19 and I took each of these data values and I subtracted 19. So this is like the X I minus X bar and then I square them. So I did that for all all those And then I went ahead and added those up. So some of C1 through C11. So 1912 is the denominator here. Is that X I minus X. Bar. Alright, so over 19 12 and then X star minus X. Bar. Well that's the the X. Star is my value that I chose was 30 and then the mean was 19 and I square that. All right. So whenever you simplify that you should get 14 897. So that's this important number right here. And that's really the hardest thing to find. So to one is the point estimate Plus or -2.26-2 Times This area here. 14 897. So that gives you 2 1 plus or minus 33.7. Almost exactly if you save the rounding to the very end And whenever you take 21 -33.7 you get 167.3 than to one plus 33.7 you get 2 34.7. Okay so that's my 95% confidence interval of the estimates. All right so now let's go on to the prediction interval. So now we have a new line but we don't know what it is. Well the critical value is still going to be the same. It's still that 2.26-2. But the standard error of the prediction is a little bit different. So it's still that 37 9372 but this time it's times the square root of one plus 1/11 plus the same thing. So it's um Whatever it was 30 -19 Squared over the 1912. So difference there it's just the one instead of one over n so it's gonna be a little bit bigger than air because we don't know as much. So 40 0.7 57 is that air? So it's quite a bit bigger. So the Point estimates still uh 201. And then the critical value is still 2.26-2. And then we're gonna multiply that by 40.757. So you get to one plus or minus 92.2 Which whenever you take 2, -92.2 you get 108.8 and then to a one plus 92 2 you get to 93 .2. So that is our prediction interval. And then the last part it asks us, well what's the difference between B&C. The prediction interval? So part D. The prediction interval is much wider and that's always going to be the case. We have those bands do to larger margin of air and also, you know, less is known as a new new equations unless is known about that equation. So there's more variability in it

View More Answers From This Book

Find Another Textbook

The data from exercise 2 follow.$$\begin{array}{r|ccccc}{x_{i}} &…

In exercise $1,$ the following estimated regression equation based on 10 obs…

Refer to exercise $21,$ where data on the production volume $x$ and total co…

The following data are the monthly salaries $y$ and the grade point averages…

Refer to the data presented in exercise $2 .$ The estimated regression equat…

A mathematical model that represents the data is $f(x)=\left\{\begin{array}…

(a) Use the method described in Exercise 45 to find the equation of the regr…

For Exercises 103–107, assume that a linear equation models each situation.<…

Find the regression equation for predicting final score from midterm score, …

The following problem is reprinted from the November 1989 Actuarial Examinat…

For the following exercises, use each set of data to calculate the regressio…

Driving speed and fuel consumption Exercise 9 (page 159$)$ gives data on the…

Let's try the re-expressed variable Fuel Consumption ( gal $/ 100 \math…

For the following exercises, refer to Table 12.$$\begin{array}{ccccccccc…

For the data in Exercises 14 in Sections $10-1$ and $10-2$ and 16 in Section…

Fitting a Line to Data In Exercises $79-82$ , find the least squares regress…

The following data were used in a regression study.$\begin{array}{cccccc…

01:10

Consider a regression study involving a dependent variable $y,$ a quantitati…

01:07

A more accurate approximation to $E\left[h\left(X_{1}, \ldots, X_{n}\right)\…

05:24

since 2000 , the Toyota Camry, Honda Accord, and Ford Taurus have been the t…

11:56

The data from exercise 2 follow.$$\frac{x_{i}|3 \quad 12 \quad 6 \quad 2…

06:45

One of the biggest changes in higher education in recent years has been the …

05:52

In exercise 18 , the data on grade point average $x$ and monthly salary $y$ …

01:42

Seven percent of mutual fund investors rate corporate stocks "very safe…

08:41

A Pew Research Center survey asked respondents if they would rather live in …

06:12

The Wall Street Joumal Subscriber Study showed data on the employment status…