Question

IN THE BELOW REGRESSION MODEL, FULLY INTERPRET THE REGRESSION (SLOPE) COEFFICIENTS AND COMMENT ON THEIR STATISTICAL SIGNIFICANCE. IN DISCUSSING STATISTICAL SIGNIFICANCE OF A REGRESSION COEFFICIENT, YOU HAVE TO JUSTIFY YOUR CHOICE OF A ONE OR TWO TAIL TEST.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.48457333
R Square	0.23481131
Adjusted R Square	0.21365402
Standard Error	1.18083028
Observations	224
ANOVA
	df	SS	MS	F	Significance F
Regression	6	92.8506974	15.4751162	11.0983638	8.6676E-11
Residual	217	302.576153	1.39436015
Total	223	395.42685
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	1.78541529	0.62754642	2.84507286	0.487%	0.54854872	3.02228186
HS_SCI	0.08346071	0.06547176	1.27475906	20.376%	-0.0455813	0.21250268
HS_ENG	0.03687216	0.07213185	0.5111773	60.975%	-0.1052966	0.17904088
HS_MATH	0.24431546	0.06254922	3.90597128	0.013%	0.12103368	0.36759724
PARENT EDUCATION (S=1)	-0.3635332	0.18775168	-1.9362446	5.414%	-0.7335835	0.00651717
PARENT EDUCATION (P=1)	0.16298646	0.24002232	0.6790471	49.783%	-0.310087	0.63605995
GENDER (M=1)	-0.108726	0.17944034	-0.6059173	54.520%	-0.4623951	0.24494307

Answer 1

From the regression output, before interpreting the regression coefficients, let us find the individual coefficient's which are significant and then interpret them.

We will use t test to interpret the significance of individual regression coefficients. Since we are testing whether slope coefficient is equal to 0 or not equal to 0, it is two tailed test.

HYPOTHESIS:

The hypothesis is given by,

$H_{0}:\beta _{j}=0$

$H_{1}:\beta _{j}\neq 0$

TEST STATISTIC:

The test statistic is given by,

$t=\beta _{j}_{hat}/Standard\, \, error(\beta _{j}_{hat})$

CRITICAL VALUE:

Since this is a two tailed test, the t critical value with (n-1) degrees of freedom at specified level of significance (5%) is denoted as $t^{*}$ .

DECISION RULE:

Thus we reject $H_{0}$ ​ , if calculated t statistic value is greater than critical value $t^{*}$ and if the p value is less than or equal to specified significance level (5%).

From the regression output,

The p value for intercept is % , HS_SCI is % , HS_ENG is
60,975
% , HS_MATH is % , PARENT EDUCATION (S=1)​ is ​% , PARENT EDUCATION (P=1​) is % ​ and GENDER (M=1)​ is % ​.

From the above p-values, we could see that the intercept, the variables HS_MATH​, PARENT EDUCATION (S=1​) are found to be statistically significant variables since the p values are less than or equal to $\alpha =0.05$ . Other variables are not statistically significant.

Now let us interpret the regression coefficients of these significant variables.

Since the intercept value is 1.78541529​. Thus the mean value of dependent variable without involving the independent variables (HS_SCI, HS_ENG, HS_MATH, PARENT EDUCATION (S=1), PARENT EDUCATION (P=1), GENDER (M=1)) is 1.785​.

The slope coefficient for the variable HS_MATH​ is 0.24431546. Thus the mean dependent variable value is 0.244 times more for HS_MATH​​ when compared to the reference category (base category).

The slope coefficient for the variable PARENT EDUCATION (S=1​) is -0.3635332​. Thus the mean dependent variable value is 0.364 times less for PARENT EDUCATION (S=1​) when compared to the reference category (base category).

F TEST:

We will use F test to interpret the overall significance of regression coefficients (model).

HYPOTHESIS:

The hypothesis is given by,

$H_{0}:\beta _{1}=\beta _{2}=..=\beta _{j}=0$

$H_{1}:Atleast\, one \, \beta _{j}\neq 0$

TEST STATISTIC:

The test statistic is given by,

F = Mean Sum of Squares of Regression / Mean Sum of Squares of Error

with (regression degrees of freedom, error degrees of freedom​)

CRITICAL VALUE:

Since this is a two tailed test, the F critical value with regression (numerator) degrees of freedom and error (denominator) degrees of freedom [6,217] at specified level of significance (5%) is denoted as $F^{*}$ ​.

DECISION RULE:

Thus we reject $H_{0}$ ​, if calculated F statistic value is greater than critical value $F^{*}$ and if the p value is less than or equal to specified significance level (5%).

From the regression output, since the p value for F test is 8.6676E-11​ which is less than specified level of significance $(\alpha =0.05)$ ​. we conclude that the overall model is significant.

R SQUARE:

The coefficient of determination (R²) is 0.23481131​ and adjusted R² is 0.21365402.

The key difference between R² and adjusted R² is that R² increases automatically as you add new independent variables to a regression equation (even if they don’t contribute any new explanatory power to the equation). Adjusted R² increases only when you add new independent variables that do increase the explanatory power of the multiple regression equation, making it a much more useful measure of how well a multiple regression equation fits the sample data than R². Thus Adjusted R² is preferred.

We should interpret Adjusted R²​ (0.21365402). Thus 21% of total variation in dependent variable is explained by the independent variables.