An engineer wants to determine how the weight of a? car, x, affects gas? mileage
ID: 3366547 • Letter: A
Question
An engineer wants to determine how the weight of a? car, x, affects gas? mileage, y. The following data represent the weights of various cars and their miles per gallon.
Car
A
B
C
D
E
Weight? (pounds), x
25452545
31003100
34103410
37703770
40404040
Miles per? Gallon, y
25.225.2
23.923.9
19.619.6
18.518.5
14.314.3
?(a) Find the? least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable.
Write the equation for the? least-squares regression line.
ModifyingAbove y with caretyequals=nothingxplus+nothing
?(Round the x coefficient to five decimal places as needed. Round the constant to one decimal place as? needed.)
Explanation / Answer
Solution
Given Miles per? Gallon as y and Weight? (in pounds) of ca as x, the equation for the? least-squares regression line is: ycap = 44.5 - 0.00717x ANSWER
[Details are given below for better understanding.]
Back-up Theory
The linear regression model Y = ?0 + ?1X + ?, ………………………………………..(1)
where ? is the error term, which is assumed to be Normally distributed with mean 0 and variance ?2.
Estimated Regression of Y on X is given by: Y = ?0cap + ?1capX, …………………….(2)
where ?1cap = Sxy/Sxx and ?0cap = Ybar – ?1cap.Xbar..………………………………..(3)
Mean X = Xbar = (1/n)sum of xi ………………………………………….………….(4)
Mean Y = Ybar = (1/n)sum of yi ………………………………………….………….(5)
Sxx = sum of (xi – Xbar)2 ………………………………………………..…………..(6)
Syy = sum of (yi – Ybar)2 ……………………………………………..……………..(7)
Sxy = sum of {(xi – Xbar)(yi – Ybar)} ………………………………………………(8)
All above sums are over i = 1, 2, …., n,n = sample size ……………………………(9)
Data
i
xi
yi
1
2545
25.2
2
3100
23.9
3
3410
19.5
4
3770
18.5
5
4040
14.3
Summary of Excel Calculations
n
5
Xbar
3373.00
ybar
20.28
Sxx
1363980
Syy
76.848
Sxy
-9786.2
?1cap
-0.0071747
?0cap
44.4803934
i
xi
yi
1
2545
25.2
2
3100
23.9
3
3410
19.5
4
3770
18.5
5
4040
14.3