Question

1. Write the regression equation.

2. Discuss the statistical significance of the model using the appropriate regression statistic at a 95% level of confidence.
3. Discuss the statistical significance of the coefficient for the independent variable using the appropriate regression statistic at a 95% level of confidence.
4. Interpret the coefficient for the independent variable.
5. What percentage of the observed variation in income is explained by the model?
6. Predict the value of a person’s income with 3 children, using this regression model.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.039342
R Square	0.001548
Adjusted R Square	0.000672
Standard Error	1.554036
Observations	1142
ANOVA
	df	SS	MS	F	Significance F
Regression	1	4.267905	4.267905	1.767229	0.183991
Residual	1140	2753.131	2.415027
Total	1141	2757.398
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.621275	0.067736	23.93504	6.6E-103	1.488373	1.754177	1.488373	1.754177
X Variable 1	1.64E-06	1.23E-06	1.329372	0.183991	-7.8E-07	4.06E-06	-7.8E-07	4.06E-06

Answer 1

(a)

The sample regression equation to estimate the dependent variable income regressed on the independent variable children (x1)

... (1)

From the given summary output

Substituting the parameter estimates in equation (1), the regression equation is given as

... (I)

(b)

The statistical significance of the model is inferred from F- statistic and the corresponding p-value of the F-statistic.

We test the hypothesis

vs

We reject Null hypothesis if p-value of the corresponding statistic is <= alpha (level of significance = (1 - level of confidence))

Here from the output, consider the p-value of the F-statistic = 0.183991, ...(2)

At 95% level of confidence, alpha = 1 - 95% = 1 - (95/100) = 1 - 0.95 = 0.05 .. (3)

From (2) and (3), p-value (0.183991) > alpha (0.05) we do not reject the null hypothesis i.e. the independent variable, children (x1) considered in the model is not statistically significant in explaining the independent variable, income (y)

(c)

The statistical significance of coefficient for the independent variable, income (x1) is inferred from t- statistic and the corresponding p-value of the t-statistic.

We test the hypothesis

vs

We reject Null hypothesis if p-value of the corresponding statistic is <= alpha (level of significance = (1 - level of confidence))

Here from the output, consider the p-value of the t-statistic = 0.183991, ...(2)

At 95% level of confidence, alpha = 1 - 95% = 1 - (95/100) = 1 - 0.95 = 0.05 .. (3)

From (2) and (3), p-value (0.183991) > alpha (0.05) we do not reject the null hypothesis i.e. the independent variable , children (x1) considered in the model is not statistically significant in predicting the independent variable, income (y)

(d) Interpret the coefficient for the independent variable.

From (I), the coefficient of the independent variable, children (x1) is interpreted as, for a unit increase in the independent variable (x1) results in an increase in the average of the dependent variable, income (y) by 1.64E^-06 (= 0.00000164)

(e)

Coefficient of determination (R Square), determines the percentage of the observed variation in income (y) is explained by the model.

From the given output,  R Square = 0.001548 (in percentage terms, 0.001548 * 100 = 0.1548%) i.e, only 0.1548% of the total variation in dependent variable income (y) is explained by the independent variable, children (x1)

(f)

Substituting the value of the independent variable, children (x1) = 3 in equation (I)

Hence the estimated value of Income, for 3 children is 1.6213 (approximately)