Question

This problem requires the use of R-Studio.

Consider the mtcars data. In R, you can use the following code to get the data:

dta <- mtcars

Use ?mtcars to read the information about this data set. In what follows, we will fit the regression model:

mpg_i=β₀ + β₀vs_i+ ε_i, i=1,2,...,n.

Note that, vs is categorical variable, whose value is 1 if the observed car has the V-shaped engine or 0 otherwise.

(A) Which of the following is the most accurate interpretation of β₀?

a) The mean of mpg when cars dont have the V-shaped engine.

b) The mean difference in mpg comparing cars having the V-shaped engine to those not having the V-shaped engine.    

c) The change in mpg that is caused by changes in vs.

d) The mean of mpg when cars have the V-shaped engine.

(B) Which of the following is the most accurate interpretation of β₁?

a) The change in mpg that is caused by changes in vs.

b) The mean of mpg when cars dont have the V-shaped engine.    

c) The mean difference in mpg comparing cars having the V-shaped engine to those not having the V-shaped engine.

d) The mean of mpg when cars have the V-shaped engine.

(C) What are the estimates of What are the estimates of β₀ and β₁? Answer to three significant figures.

(D) What is the p-value for the null hypothesis that the coefficient of Engine type (vs) equals 0. Is it significant? What does it mean?

a) The p-value is 8.85e−16 and thus it is significantly small. It indicates that the mean of mpg for the second group (vs = 1, straight) is larger than that for the second group (vs = 0, V-shaped).

b) The p-value is 3.42e−5 and thus it is significantly small. It indicates that the mean of mpg for the second group (vs = 1, straight) is smaller than that for the second group (vs = 0, V-shaped).    

c) The p-value is 8.85e − 16 and thus it is significantly small. It indicates that the mean of mpg for the second group (vs = 1, straight) is smaller than that for the second group (vs = 0, V-shaped).

d) The p-value is 3.42e−5 and thus it is significantly small. It indicates that the mean of mpg for the second group (vs = 1, straight) is larger than that for the second group (vs = 0, V-shaped).

(E) What will be the fitted value when vs = 0 and vs = 1 respectively?

a) 16.62, 7.94

b) 7.94, 16.62    

c) 16.62, 24.56

d) 0, 16.62

Answer 1

According to Concept of intercept in Regression

Answer of A is

a) The mean of mpg when cars dont have the V-shaped engine.

B)

β1 - This is the SLOPE of the regression line. Thus this is the amount that the Y variable (dependent) will change for each 1 unit change in the X variable.

But Here we have X as a Categorical variable... so answer is

c) The mean difference in mpg comparing cars having the V-shaped engine to those not having the V-shaped engine.

C) estimates of β0 and β1

Intercept(β0) --- 16.617
vs(β1) --- 7.940

D)

d) The p-value is 3.42e−5 and thus it is significantly small. It indicates that the mean of mpg for the second group (vs = 1, straight) is larger than that for the second group (vs = 0, V-shaped).

E)

fitted value when vs = 1

Fitted Value=16.617+7.94*(1)=24.56

fitted value when vs = 0

Fitted Value=16.617+7.94*(0)=16.62

So option c) 16.62, 24.56 is Correct

All the Best