Question

A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.

SUMMARY OUTPUT

Regression Statistics
Multiple R	0.865
R Square	0.748
Adjusted R Square	_____
Standard Error	5.195
Observations	50

ANOVA

	df	SS	MS	F	Significance F
Regression	____	3605.7736	1201.925	_____	0.0001
Residual	____	1214.2264	_____
Total	49	4820.0000

	Coeff.	St. Error	t Stat	P-value
Intercept	-1.6335	______	-0.281	0.7798
Family Income	0.4485	0.1137	_____	0.0003
Family Size	_____	0.8062	5.286	0.0001
Education	-0.6517	0.4319	-1.509	0.1383

1. Complete the above Summary Output by filling empty spaces. (8 marks)

2. What percentage of the variability in house size is explained by this model? (1 mark)

3. Write a regression equation by using values given in the Tables above. (1.5 mark)

4. What is the predicted house size for an individual earning an annual income of $40,000, having a family size of 4, and having 13 years of education? (1.5 mark)

5. Test whether the whole regression model is significant? (3 marks)

Answer 1

Multiple R	0.865
R Square	0.748
Adjusted R Square	__0.7315___
Standard Error	5.195
Observations	50

ANOVA

	df	SS	MS	F	Significance F
Regression	__3__	3605.7736	1201.925	__45.6091___	0.0001
Residual	__46__	1214.2264	__26.35274___
Total	49	4820.0000

	Coeff.	St. Error	t Stat	P-value
Intercept	-1.6335	__5.8131____	-0.281	0.7798
Family Income	0.4485	0.1137	__3.9445___	0.0003
Family Size	__4.2615___	0.8062	5.286	0.0001
Education	-0.6517	0.4319	-1.509	0.1383

the steps for filling this table is shown below

sample size n, is 50

number of predictors ,p =3

$\small adjusted R^{2} =1-\frac{(1-R^{2})(n-1)}{(n-p-1)}$

                   = $\small 1-\frac{(1-0.748)(50-1)}{(50-3-1)}$

                    =

1- 0.252 x 49 46

                = 0.7315

degrees of freeom for sum of sqaures of total is n-1

degrees of freeom for sum of sqaures of regression is p

here p=3 [to check if the answer is correct , see if SS_Reg /3 is the given MS_{Reg ]}

degrees of freeom for sum of sqaures of residuals is n-p-1

n-p-1 =50-3-1 = 46

$\small MS_{Res}=\frac{SS_{Res}}{n-p-1} = \frac{1212.2264}{46}$

= 26.35274

F statistic is used to test the overall significance of regression

$\small F =\frac{MS_{Reg}}{MS_{Res}}$

      $\small =\frac{1201.925 }{26.35274}$    = 45.6091

t statisitic in the table is used to test the significance of each coefficients. with null hypothesis as $\small \beta _{i}=0$ as alternatice hypothesis :$\small \beta _{i}\neq 0$

where $\small \beta _{i}$ is the regression coefficient of ith   variable.

$\small t =\frac{\widehat{\beta _{i}}}{se(\widehat{\beta _{i}})}$  

standard error of intercept , $\small se(\widehat{\beta _{0}}) =\frac{\widehat{\beta _{0}}}{t}$

                                                           $\small = \frac{-1.6335}{-0.281}$       = 5.8131

t statistic for family income,   

$\small t=\frac{\widehat{\beta _{1}}}{se(\widehat{\beta _{1}})}$ = $\small =\frac{0.4485}{0.1137}$ = 3.9445

coefficient for family size , $\small \widehat{\beta _{2}} =t\times se(\widehat{\beta _{2}})$

                                                =0.8062*5.286 = 4.2615

$\small \bullet$ What percentage of the variability in house size is explained by this model?

The coefficient of determination, denoted R² , is the proportion of the variance in the dependent variable that is predictable from the independent variable .

The value of R squared is given ,0.748.

Therefore 74.8% of the variability in house size is explained by this model

$\small \bullet$ Write a regression equation by using values given in the Tables above .

Let Y be the house size. Let X₁, X₂, X₃ represents family income , family size, education respectively. and $\small \beta _{1},\beta _{2} , \beta _{3}$ represents their coeffient respectively. $\small \beta _{0}$ be the intercept.

regression equation is

$\small \widehat{Y}= \widehat{\beta _{0}}+\widehat{\beta _{_{1}}}X_{1}+\widehat{\beta _{_{2}}}X_{2}+\widehat{\beta _{_{3}}}X_{3}$

$\small \widehat{Y}=-1.6335+0.4485X_{1}+4.2615X_{2}-0.6517X_{3}$

$\small Y=-1.6335+0.4485X_{1}+4.2615X_{2}-0.6517X_{3}+\varepsilon$

$\small \bullet$ What is the predicted house size for an individual earning an annual income of $40,000, having a family size of 4, and having 13 years of education?

$\small \widehat{Y}=-1.6335+0.4485\times 40000+4.2615\times 4-0.6517\times 13$

       =19546.9404

$\small \bullet$ Test whether the whole regression model is significant?

We use Fstatisitic for over all significance of regression model.

H₀: All coefficients equals zero

H₁: atleat one of the coefficient is not equal to zero.

It is already calculated. p value associated with F statisitic is given in the same table right to it in the column significance of F.

p value =0.0001

if we take $\small \alpha$ = 0.05 or 0.01, or 0.001

p value < $\small \alpha$

null hypothesis is rejected , which means atleast one of the coefficient is not equal to zero. .Therefore the regression is over all significant.

Now we can also check significance of individual coefficients with t statistic, with (say) 0.05 level of significance.

p value is given right to t statistic.

we can see p value for family income and family size is less than level of significance 0.05.

but p value for education happens to be greater than 0.05

Therefore we can doubt the significance of education.

To check the if whole regression is significant , Ftest will be enough.