Question

1.

Following R output is from fitting multiple regression of House price (Y) in thousand based on Finish Area(X1), Age(X2) and Bedroom(X3).

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 190.11 2.587 73.48 0.00866 **
X1 2.30 0.010 228.00 0.00279 **
X2 -2.46 0.064 -38.03 0 .01674 *
X3 36.67 0.359 101.96 0.00624 **
-----------------------------
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9204 on 1 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 7.813e+04 on 3 and 1 DF, p-value: 0.0026

Estimate the price of the house in thousand when area=150, age=25 and bedroom=4.

2.

Following R output is from fitting multiple regression of House price (Y) in thousand based on Finish Area(X1), Age(X2) and Bedroom(X3).

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 190.11 2.587 73.48 0.00866 **
X1 2.30 0.010 228.00 0.00279 **
X2 -2.46 0.064 -38.03 0 .01674 *
X3 36.67 0.359 101.96 0.00624 **
-----------------------------
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9204 on 1 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 7.813e+04 on 3 and 1 DF, p-value: 0.00263

Find the p-value to test the hypotheses H0:β2=0  vs  H1:β2≠0

3.

Following R output is from fitting multiple regression of House price (Y) in thousand based on Finish Area(X1), Age(X2) and Bedroom(X3).

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 190.11 2.587 73.48 0.00866 **
X1 2.30 0.010 228.00 0.00279 **
X2 -2.46 0.064 -38.03 0 .01674 *
X3 36.67 0.359 101.96 0.00624 **
-----------------------------
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9204 on 1 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 7.813e+04 on 3 and 1 DF, p-value: 0.0026

Test the hypothesis  H0:β2=0  vs  H1:β2≠0   H0:β2=0  vs  H1:β2≠0   α=1%

4.

Following R output is from fitting multiple regression of House price (Y) in thousand based on Finish Area(X1), Age(X2) and Bedroom(X3).

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 190.11 2.587 73.48 0.00866 **
X1 2.30 0.010 228.00 0.00279 **
X2 -2.46 0.064 -38.03 0 .01674 *
X3 36.67 0.359 101.96 0.00624 **
-----------------------------
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9204 on 1 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 7.813e+04 on 3 and 1 DF, p-value: 0.00263

Find the p-value to test the hypotheses:

H0:β1=β2=β3=0H1:βi≠0 for at least on i

Answer 1

Question (1)

The regression equation is

Price = 190.11 + 2.3 * Area + (-2.46) * Age + 36.67 * Bedroom

Given area=150, age=25 and bedroom=4

Prcie = 190.11 + 2.3 * 150 + (-2.46) * 25 + 36.67 * 4

= 620.29

Price of the house in thousand when area=150, age=25 and bedroom=4 is 620.29

Question (2)

the p-value to test the hypotheses H0:β2=0  vs  H1:β2≠0 is the the p-value for X2 variable which is 0.01674

Question (3)

If the p-value is less than the significance level, then the Null Hypothesis will be rejected, else we fail to reject the Null Hypothesis

Here H0:β2=0  vs  H1:β2≠0

We need to look at the p-value for X2 variable

The p-value for X2 variable is 0.01674

The Significance level is 0.01 since there is one * after the p-value

Here the p-value of 0.01674 is more than The Significance level of 0.01, Hence we fail to reject the Null Hypothesis or H0

So β2=0

Question (4)

Find the p-value to test the hypotheses:

H0:β1=β2=β3=0H1:βi≠0 for at least on i

The p-value for all the β combined will be the p-value of the F-statistic

The p-value for F-statistic here is 0.00263