Problem

In exercise loan estimated regression equation wa…

05:31

Problem 25 Easy Difficulty

$\begin{array}{l}{\text { Small cars offer higher fuel efficiency, are easy to maneuver and park, and have generally }} \\ {\text { performed well in road tests. The following data show the overall fuel efficiency rating (miles }} \\ {\text { per gallon), predicted reliability (highest }=5 \text { , safety rating (highest }=5 ) \text { , and the price (S) for }} \\ {20 \text { small cars (Consumer Reports, March } 2009 ) \text { . }}\end{array}$
$\begin{array}{l}{\text { a. Determine the estimated regression equation that can be used to price of a }} \\ {\text { small car given its overall fuel efficiency rating, predicted reliability, and safety rating. }} \\ {\text { b. Use the } F \text { test to determine the overall significance of the relationship. What is the }} \\ {\text { conclusion at the } .05 \text { level of significance? }}\end{array}$
$\begin{array}{l}{\text { c. Use the } t \text { test to determine the significance of each independent variable. What is your }} \\ {\text { conclusion at the } .05 \text { level of significance? }}\end{array}$
$\begin{array}{l}{\text { d. Remove any independent variable that is not significant from the estimated regression }} \\ {\text { equation. What is your recommended estimated regression equation? Compare the }} \\ {\text { value of } R^{2} \text { for this estimated regression equation with the value of } R^{2} \text { for the estimated }} \\ {\text { regression equation from part (a). Discuss the differences. }}\end{array}$

Answer

a) $\hat{y=-18861.2586-281.6487 x_{1}+207.5047 x_{2}+1700.4603 x_{3}}$
b) 13.8765
c) $-3.7652$
1.2062

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Intro Stats / AP Statistics

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Chapter 13

Multiple Regression

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Video Transcript

So this question we are given some data about small cars and were basically given their MPG reliability and safety. So these are the three X variables and were given the Y variable, which is price, and were asked to determine the multiple regression equation four price. So let's go ahead and do that. So our input wide range is basically price. So that goes up until to sell 21. We have he want to If 21 here for our input Wire ridge for our input X range. We start at B one and we will end at B 21 because we have the three cold P, C and D keep our confidence level at 99% are output range. We are going to put it over here, so we have it at 8. 24. We make sure we have our wide range and our X range our output and we click okay to get our multiple linear regression equation. So we have all of our regression statistics off here. So we see our r squared 0.72 which is all right then going on. So we have our coefficients here for the three values. So basically, let's use thes coefficients on Get Our multiple in Your regression equations are first. It's for mpg and reliability and then safety. So right me close to intercept minus 2 81 and C ci for us to or seven or the liability yes has been 700 times u p we have prices equals two minus check. Our intercept crisis equals to 18861 minus to 81 times x one plus 27 times x two plus 1700 times extreme. So we basically have power equation over here. Yes, we're part is now let's move on to Part B now in part, we were asked to use the F test to determine the overall significance of the relationship. So we have our f statistic over here. So I zahra a statistic r p value this 0.41 So that's pretty small. Mhm. And so since the p value off point or one is less than 0.45 relationship, they significant and one or more off the variables. Uh huh. Not people to thio. There is a significant relationship between why and at least one or more off the variables, which is what part of being and we have. Statistics on the P value is telling us now moving on to part Seaver as to use the team tests to determine the significance off each variable. So let's do that. So we have each variables. We have mpg reliability and safety and were asked to use the teachers to determine the significance of each of these variables at 0.45 So we see that mpg and safety are less than 0.5 and so therefore they're significant. But reliability is not significant, so let's go ahead and starting. So for MPT, you know you just left standpoint or five. Is this significant for reliability? I love you. Daddy needs into four, which is more than point or fight. So it's not significant. And then, for our last variable safety, very small pre values. Please, Please Do you have a new leach to time standard about a minus bi sport, which speaks. Can you suggest that and soup? We have our answer to part C to variable significant one. Not significant now for party, we are told toe first to remove any independent variable that is not significant from the equation. So let's do that. Okay, so we're going to have to remove reliability from the equation. So let's go ahead and do that first. So now we have re motors liability from power equation, and that's two linear regression. So we have the wide range, the X Range 99% and we get our summary output here. So we see we see our r squared and there we have it. So we have our coefficients for the intercept mpg and 60. So let's write our linear regression equations. We have 25 deep crisis people's through the intercept when we have some PG and safety until there's some differences. Flight differences When politicians between the party and part team, they're pretty similar and then were asked in 14 after writing the recommended regression equation to compare the R squared 44 so are our squared for party was 0.722 adjusted. Our square was 0.67 So it's right that right now are square or party with just two variables on adjusted R squared 0.69 point 7.66 So we can basically conclude that our differences in our square were very I mean, we're very minimal or very little even after removed after removing non significant variable so we can include that the variable that's not count for much off variability in the data and for two significant variables account for most off variability. So basically, because they r squared values were pretty similar or even when we removed the non significant variables, we basically can see that the two significant variables are what account for most of the variability in the data. And the non significant variable really does not contribute much to the equation. So that's our conclusion, and this is the answer for 25 party through people.

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