Chapter 12.4, Problem 8E

To determine

The multiple regression equation and define each of the variables used in the multiple regression equation. Also check any one of the explanatory variables be eliminated from the model and mention it.

Answer to Problem 8E

Solution:

The multiple regression equation for the given table is $\widehat{\mathrm{y}}=-0.616+0.975{\mathrm{x}}_1+0.015{\mathrm{x}}_2$ with the y-intercept, ${\mathrm{ꞵ}}_0=-0.616$, the coefficient of the first explanatory variable, college freshman’s high school GPAs, ${\mathrm{ꞵ}}_1=0.975$ and the coefficient of the second explanatory variable, college freshman’s ACT scores, ${\mathrm{ꞵ}}_2=0.015$ and also it needs to eliminate the second explanatory variable, college freshman’s ACT scores, as it is not statistically significant.

Explanation of Solution

Formula Used:

Multiple Regression Model:

A multiple regression model is a linear regression model using two or more explanatory variable to predict a response variable, given by

$\widehat{\mathrm{y}}={\mathrm{b}}_0+{\mathrm{b}}_1{\mathrm{x}}_1+{\mathrm{b}}_2{\mathrm{x}}_2+\cdots +{\mathrm{b}}_{\mathrm{k}}{\mathrm{x}}_{\mathrm{k}}$

Where ${\mathrm{x}}_1,\mathrm{ }{\mathrm{x}}_2,\mathrm{ }\cdots ,\mathrm{ }{\mathrm{x}}_{\mathrm{k}}$ are the explanatory variables in the model and ${\mathrm{b}}_1,\mathrm{ }{\mathrm{b}}_2,\mathrm{ }\cdots ,\mathrm{ }{\mathrm{b}}_{\mathrm{k}}$, of the explanatory variables are the sample estimates of the corresponding population parameters ${\mathrm{ꞵ}}_1,\mathrm{ }{\mathrm{ꞵ}}_2,\mathrm{ }\cdots ,\mathrm{ }{\mathrm{ꞵ}}_{\mathrm{k}}$. As before, the $\mathrm{y}$-intercept of the multiple regression equation is ${\mathrm{b}}_0$, which is the sample estimate of the population parameter, ${\mathrm{ꞵ}}_0$.

Null and Alternative Hypothesis for an ANOVA test:

The null and alternative hypotheses for an ANOVA test to analyse the statistical significance of the linear relationship between the variables in a multiple regression model as follows.

${\mathrm{H}}_0:\mathrm{ }{\mathrm{ꞵ}}_1={\mathrm{ꞵ}}_2=\cdots ={\mathrm{ꞵ}}_{\mathrm{k}}=0$

${\mathrm{H}}_{\mathrm{a}}:$ At least one coefficient does not equal 0.

${\mathrm{ꞵ}}_1,\mathrm{ }{\mathrm{ꞵ}}_2,\mathrm{ }\cdots ,{\mathrm{ꞵ}}_{\mathrm{k}}$ are the coefficients of the explanatory variables, and

$\mathrm{k}$ is the number of explanatory variables in the model.

Conclusions Using $\mathrm{p}$-value:

If $\mathrm{p}$-value $\le \mathrm{{\rm A}}$, then reject the null hypothesis.

If $\mathrm{p}$-value $>\mathrm{{\rm A}}$, then fail to reject the null hypothesis.

The ANOVA table given as follows:

	$\mathrm{C}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$	$\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{ }\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}$	$\mathrm{t}\mathrm{ }\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}$	$\mathrm{P}-\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}$
Intercept	-0.616112354	0.345507101	-1.783211841	0.092415337
High School GPA	0.975068267	0.186279465	5.234437779	6.7297E-05
ACT Score	0.015000337	0.019048101	0.787497806	0.441832554

The block of information gives the coefficients of the multiple regression equation and the coefficients of the explanatory variables.

The first row of the coefficient column gives the predicted value for the y-intercept, ${\mathrm{ꞵ}}_0=-0.616$. The second row of that column gives the predicted value for the coefficient of the first explanatory variable, freshman’s high school GPAs, ${\mathrm{ꞵ}}_1=0.975$. The third row of that column gives the predicted value for the coefficient of the second explanatory variable, freshman’s ACT scores, ${\mathrm{ꞵ}}_2=0.015$.

Substituting the estimated coefficients into the multiple regression model, we have the following equation for predicting the college freshman GPA after their first year based on their high school GPAs, ${\mathrm{x}}_1$, and their ACT scores, ${\mathrm{x}}_2$.

$\widehat{\mathrm{y}}=-0.616+0.975{\mathrm{x}}_1+0.015{\mathrm{x}}_2$

Let’s consider the individual explanatory variables more closely. Look at the $\mathrm{p}$-values for the explanatory variables.

$\mathrm{P}-\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}$

0.092415337

6.7297E-05

0.441832554

The $\mathrm{p}$-values test the null hypothesis that the coefficient of a particular explanatory variable equals 0.

The null and alternative hypotheses for the first explanatory variable would be as follows.

${\mathrm{H}}_0:\mathrm{ }{\mathrm{ꞵ}}_1=0$

$H_a: {\mathrm{ꞵ}}_1\ne 0$

The null and alternative hypotheses for the other explanatory variables are similar. A small $\mathrm{p}$-value (such as one less than 0.05) indicates that there is sufficient evidence to support the claim that the coefficient is not 0, and therefore the linear relationship between this particular variable and the response variable is statistically significant.

If the $\mathrm{p}$-value is not small (for instance, those greater than 0.05) then this particular variable may not be useful in predicting the value of the response variable.

In this case of predicting the college freshman GPA after their first year, notice that the $\mathrm{p}$-values for the explanatory variables of college freshman’s high school GPAs is approximately 6.7297E-05, that is 0.0000673 which is smaller than 0.05. This indicates that these explanatory variables are very likely to be effective in predicting the response variable.

Similarly, the $\mathrm{p}$-values for the explanatory variable of college freshman’s ACT scores is approximately 0.4418, which is much higher than 0.05.

Thus, including freshman’s ACT scores in the multiple regression model may not be useful. Need to recalculate the multiple regression model without this variable. However, that the $\mathrm{p}$-values calculated in the ANOVA table measure the influence of each explanatory variable with all other variables taken.

Therefore, it needs to eliminate the second explanatory variable, college freshman’s ACT scores, as it is not statistically significant.

Final Statement:

The multiple regression equation for the given table is $\widehat{\mathrm{y}}=-0.616+0.975{\mathrm{x}}_1+0.015{\mathrm{x}}_2$ with the y-intercept, ${\mathrm{ꞵ}}_0=-0.616$, the coefficient of the first explanatory variable, college freshman’s high school GPAs, ${\mathrm{ꞵ}}_1=0.975$ and the coefficient of the second explanatory variable, college freshman’s ACT scores, ${\mathrm{ꞵ}}_2=0.015$ and also it needs to eliminate the second explanatory variable, college freshman’s ACT scores, as it is not statistically significant.