Question

1) A consumer organization wants to develop a regression model to predict gasoline mileage​ (as measured by miles per​ gallon) based on the horsepower of the​car's engine and the weight of the car​ (in pounds). A sample of 20 recent car models was​ selected, with the results recorded in the accompanying table.

MPG 15.3, 19.2, 20.1, 18.5, 17.5, 27.2, 44.6, 27.2, 28.0, 21.2, 28.0, 36.1, 20.1, 29.9, 36.0, 36.4, 33.7, 32.9, 24.2, 39.3 Horsepower - 190, 102, 142, 171, 166,67,64,82,91, 136, 90, 67, 88, 60,75, 64, 74, 101, 119, 75 weight- 4757, 3538, 3209, 4449, 4292, 3192, 2111, 2489, 2605, 3875, 2673, 1805, 2964, 2375, 1975, 2948, 2212, 2611, 2935, 2071

a. State the multiple regression equation.

Est. Mean Mileage= ___+(__)+(__)Weight

​(Round to five decimal places as​ needed.)

b. Interpret the meaning of the​ slopes,bH and bW​, that you entered above

Each increase of one unit in horsepower is estimated to result in a

decrease in the mean gasoline mileage of ___​units, holding weight constant.

Each increase of one unit in weight is estimated to result in the decrease in the mean gasoline mileage of __units, holding horsepower constant

c. Explain why the regression​ coefficient,

b0​, has no practical meaning in the context of this problem.

A.

The interpretation of b0 has no practical meaning here because it would correspond to the estimated mean weight when a car has 0 gasoline mileage and 0 horsepower.

B.

The interpretation of b0 has no practical meaning here because it would correspond to the estimated mean gasoline mileage when a car has 0 horsepower and 0 weight.

C.

The interpretation of b0 has no practical meaning here because it would correspond to the estimated mean horsepower when a car has 0 gasoline mileage and 0 weight.

d. Predict the miles per gallon for a car that has 60 horsepower and weighs 2000 pounds.

The gasoline mileage​ (as measured by miles per​ gallon) prediction for a car that has 60 horsepower and weighs 2000 pounds is ____

miles per gallon.

​(Round to three decimal places as​ needed.)

2)A consumer organization wants to develop a regression model to predict gasoline​mileage​ (as measured by miles​ per​ gallon) based on the horsepower of​ the​ car's engine and the weight of the​ car​ (in​pounds). A sample of 20 recent car models​was​ selected, with the results recorded in the accompanying table.

The resulting ANOVA table is below. Complete parts​ (a) through​ (c).

Regression

Statistics

Multiple R

0.9770

R Square

0.9546

n

8

Source

Regression

Degrees of

Freedom

2

Sum of

Squares

16.94097

Mean Square

8.47049

F

52.56

p-value

0.0004

Error

Degrees of

Freedom

5

Sum of

Squares

0.80575

Mean Square

0.16115

total

Degrees of

Freedom

7

Sum of

Squares

17.74672

a. Determine whether the model is significant​ (overall) at the

0.05 level of significance. Choose the correct answer below.

A.

Do not reject H0. There is insufficient evidence to prove that at least one slope is not zero. We cannot prove that there is a linear relationship between mileage and at least one of the horsepower and weight.

B. Reject H0. There is insufficient evidence to prove that at least one slope is not zero. We cannot prove that there is a linear relationship between mileage and at least one of horsepower and weight.

C. Reject H0. There is sufficient evidence to prove that at least one slope is not zero.  There is a linear relationship between mileage and at least one of horsepower and weight.

D. Do not reject H0. There is sufficient evidence to prove that at least one slope is not zero.  There is a linear relationship between mileage and at least one of horsepower and weight.

b. What is the value coefficient of multiple​determination, R2, and interpret its meaning.

R2=

(Round to four decimal places as​needed.)

c. Interpret the meaning of R2.

It indicates that ___% of the variation in mileage in the sample can be explained by its linear relationship with horsepower and weight.

​(Round to two decimal places as​ needed.)

3)A consumer organization wants to develop a regression model to predict gasoline​mileage​ (as measured by miles​ per​ gallon) based on the horsepower of​ the​ car's engine and the weight of the​ car​ (in​pounds). A sample of 25 recent car models​was​ selected, with the results recorded in the accompanying table.

Variable Coefficient Std Error tstat ​p-value Lower​ 95%Upper​ 95%

Intercept 0.00282 0.05667 0.05 0.9608 -0.11471 0.12035

Horsepower-0.71635 0.03717 -19.27 0.0000 -0.79344 -0.63926

Weight -0.11854 0.08734 -1.36 0.1885 -0.29968 0.0626

At the 0.05 level of​ significance, determine whether each independent variable makes a significant contribution​ (individually) to the regression model.

Can we prove that there a relationship between mileage and​ horsepower?

*No.Do not reject the null. There is insufficient enough evidence to conclude that the slope of horsepower is not 0.

Yes. Reject the null. There is enough evidence to conclude that the slope of horsepower is not 0.

Can we prove that there a relationship between mileage and​ weight?

No. Do not reject the null. There is insufficient enough evidence to conclude that the slope of weight is not 0.

Yes.Reject the null. There is enough evidence to conclude that the slope of weight is not 0.

Answer 1

Regression Statistics
Multiple R	0.850589
R Square	0.723502
Adjusted R Square	0.690973
Standard Error	4.577843
Observations	20

ANOVA
	df	SS	MS	F	Significance F
Regression	2	932.2191	466.1095	22.24161	1.8E-05
Residual	17	356.2629	20.95664
Total	19	1288.482

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	51.04277	3.945965	12.93543	3.17E-10	42.71751	59.36803
Horsepower	-0.04338	0.056017	-0.77439	0.449336	-0.16157	0.074807
weight	-0.00639	0.002607	-2.45172	0.025324	-0.01189	-0.00089

a. State the multiple regression equation.

Est. Mean Mileage= 51.04277 -0.04338*Horse power - 0.00639* Weight

b. Interpret the meaning of the​ slopes,bH and bW​, that you entered above

Each increase of one unit in horsepower is estimated to result in a

decrease in the mean gasoline mileage of 0.04338 units, holding weight constant.

Each increase of one unit in weight is estimated to result in the decrease in the mean gasoline mileage of 0.00639 units, holding horsepower constant

c. Explain why the regression​ coefficient, b0​, has no practical meaning in the context of this problem.

The interpretation of b0 has no practical meaning here because it would correspond to the estimated mean gasoline mileage when a car has 0 horsepower and 0 weight.

d. Predict the miles per gallon for a car that has 60 horsepower and weighs 2000 pounds.

The gasoline mileage​ (as measured by miles per​ gallon) prediction for a car that has 60 horsepower and weighs 2000 pounds is 35.657 miles per gallon.

miles per gallon = 51.04277 -0.04338*60 - 0.00639* 2000

2)

a. Determine whether the model is significant​(overall) at the 0.05 level of significance. Choose the correct answer below.

Since p-value = 0.0004, Reject H0. There is sufficient evidence to prove that at least one slope is not zero. There is a linear relationship between mileage and at least one of horsepower and weight.

b) What is the value coefficient of multiple​determination, R2, and interpret its meaning.

R²= 0.9770

c. Interpret the meaning of R2.

It indicates that 97.70 % of the variation in mileage in the sample can be explained by its linear relationship with horsepower and weight.

3) If p-value is less than 0.05 level of​ significance, then reject null hypothesis.

a) p-value is 0.0000 hence, Yes. Reject the null. There is enough evidence to conclude that the slope of horsepower is not 0.

b) p-value is 0.188 > 0.05, No. Do not reject the null. There is insufficient enough evidence to conclude that the slope of weight is not 0.