Question

Simple Linear Regression Analysis

2. Weight of Car: Miles gallon – Do heavier cars really use more gasoline? The following data were obtained from Consumer Reports (Vol 62 no. 4). Weight of car is in hundreds of pounds.

Car Weight 27 44 32 47 23 40 34 52
MPG 30 19 24 13 29 17 21 14

A simple linear regression of the model MPG = b0 + b1 WEIGHT
The results are shown below:

MPG & CAR WEIGHT
REGRESSION FUNCTION & ANOVA FOR MPG

MPG = 43.32625 - 0.600702 WEIGHT

R-Squared = 0.895426
Adjusted R-Squared = 0.877997
Standard error of estimate = 2.236055
Number of cases used = 8


Analysis of Variance

p-value
Source SS df MS F Value Sig Prob
Regression 256.87 1 256.87 51.37567 0.000372
Residual 29.99 6 4.99
Total 286.875 7

MPG & CAR WEIGHT
REGRESSION COEFFICIENTS FOR MPG

Two-Sided p-value
Variable Coefficient Std Error t Value Sig Prob
Constant 43.32625 3.23051 13.41156 0.000011
WEIGHT -0.60070 0.08381 -7.16768 0.000372 *

Standard error of estimate = 2.236055
Durbin-Watson statistic = 0.995097

Questions:

1. What sort of relationship exists between MPG and car weight?

2. Does the relationship make sense to you? Why or why not?

3. Test the hypotheses H0: b1 = 0 against H A: b1 ?0 a level of significance ? = 0.01. What is your conclusion?

MODEL: MPG= b0 + b1 WEIGHT

H0: b1 = 0

H A: b1 ? 0

4. What is your conclusion?

Answer 1

1. According to the regression equation, the slope is negative, it implies that there is a negative relationship between both the variables that is the weight of the car and MPG. That is as the one variable increases other decreases.

Using R- Squared from the output the correlation coefficient is just the square root of R - Squared.

The correlation is negative since the slope of regression is negative.

2. Yes, the negative relationship makes the sense because as the weight of the car increases the MPG decreases. Since weight affects MPG.

3. The null and alternative hypothesis are:

H₀: There is no significant linear relationship between the weight of the car and MPG that is b₁ = 0

H1: There is significant linear relationship between the weight of the car and MPG that is b₁ is not equal to 0

T-test statistics:

The ANOVA output is already given

So the test statistics value corresponding to the slope that is corresponding to the variable weight is -7.16768

t = -7.16768

P-value = 0.000372

Alpha = level of significance = 0.01

Decision rule: If P value > alpha then fail to reject the null hypothesis otherwise reject the null hypothesis.

Here P - value is less than alpha so reject the null hypothesis.

4. Conclusion: Reject the null hypothesis that is there is a significant linear relationship between the weight of the car and MPG.