Question

Question: Forecasting a job applicant’s merit rating. A large research and development firm rates the performance of each member of its technical staff on a scale of 0 to 100, and this merit rating is used to determine the size of the person’s pay raise for the coming year. The firm’s personnel department is interested in developing a regression model to help forecast the merit rating that an applicant for a technical position will receive after being employed for 3 years. The firm proposes to use the following second-order model to forecast the merit ratings of applicants who have just completed their graduate studies and have no prior related job experience:

$\mathit{E}\left(y\right)\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{\mathit{x}}_{\mathbf{2}}^{\mathbf{2}}$, where

y = Applicant’s merit rating after 3 years

x₁= Applicant’s GPA in graduate school

x₂= Applicant’s total score (verbal plus quantitative) on the Graduate Record Examination (GRE)

The model, fit to data collected for a random sample of n=40employees, resulted in$\mathbf{SSE}\mathbf{=}\mathbf{1830}\mathbf{.}\mathbf{44}and\mathbf{SS}\left(\mathbf{model}\right)\mathbf{=}\mathbf{4911}\mathbf{.}\mathbf{5}$. The reduced model$\mathit{E}\left(y\right)\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}$ is also fit to the same data, resulting in$\mathbf{SSE}\mathbf{=}\mathbf{3197}\mathbf{.}\mathbf{16}$.

1. Identify the appropriate null and alternative hypotheses to test whether the complete (second-order) model contributes information for predicting y.
2. Conduct the test of the hypothesis given in part a. Test using. Interpret the results in the context of this problem.
3. Identify the appropriate null and alternative hypotheses to test whether the complete model contributes more information than the reduced (first-order) model for predicting y.
4. Conduct the test of the hypothesis given in part c. Test using. Interpret the results in the context of this problem.
5. Which model, if either, would you use to predict y? Explain.

Short answer

Answer

1. The null and alternate hypothesis for testing whether the second order model contributes information for the prediction of y can be written as
   $H_0:{\beta }_4={\beta }_5=0$while At least one of the parameters ${\beta }_3,{\beta }_4 \text{or} {\beta }_5$or is non-zero.
2. At a 95% significance level${\beta }_4\ne {\beta }_5\ne 0$ , the second-order model doesn’t contribute much to the model.
3. The null and alternate hypotheses for testing whether the complete model contributes more information than the reduced model for the prediction of y can be written as$H_0:{\beta }_3={\beta }_4={\beta }_5=0$ while

At least one of the parameters${\beta }_3,{\beta }_4 \text{or} {\beta }_5$.

4. At the 95% significance level, at least one of the parameters ${\beta }_3,{\beta }_4 \text{or} {\beta }_5$is non-zero.

5. Since the hypothesis testing concludes the complete model is a better fit for the data, the complete model would be more useful for predicting y.

Step-by-step solution

Given information 

The second order model used by the firm is given as $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2+{\beta }_4{x}_1^2+{\beta }_5{x}_2^2$,there are 40 observations and the,$\text{SSE}=1830.44\text{,SS}\left(\text{model}\right)=4911.5$ .

The reduced model is given where and the test needs to be conducted .

(a) Hypothesis statement

The Research and development firm is doing a regression analysis to find if certain variables affect their employees’ merit scores or not. The firm first fits a second-order equation and then fits a reduced equation with only one variable.

Here, the null and alternate hypotheses for testing whether the second order model contributes information for the prediction of y can be written as$H_0\text{:}{\beta }_4 = {\beta }_5=0$while

At least one of the parameters${\beta }_4 \text{or} {\beta }_5$ is non-zero.

(b) Hypothesis testing

$H_0\text{:}{\beta }_4= {\beta }_5=0$

H_a :At least one of the parameters${\beta }_4 \text{or} {\beta }_5$ is non-zero.

Here,

$$\begin{gathered} F-\text{test} \text{statistic}=\frac{SSE}{n-\left(k+1\right)} \\ =\frac{1830.44}{40-6} \\ =53.836 \end{gathered}$$

Value of $F_{0.05,39,39}$is 1.509.

H₀ :is rejected if $F-statistic>F_{0.05,39,39}$.

There is sufficient evidence to reject H₀at 95% confidence interval.

Therefore, ${\mathbf{\beta }}_{\mathbf{4}}\ne {\mathbf{\beta }}_{\mathbf{5}}\ne \mathbf{0}$meaning that the second order model contributes much to the model.

(c) Hypothesis statement

The null and alternate hypothesis for testing whether the complete model contributes more information than the reduced model for the prediction of y can be written as$H_0\text{:}{\beta }_3={\beta }_4={\beta }_5=0$
while At least one of the parameters ${\beta }_3,{\beta }_4 \text{or} {\beta }_5$is non-zero.

(d) Hypothesis testing

$H_0\text{:}{\beta }_3={\beta }_4={\beta }_5=0$

H_a:At least one of the parameters ${\beta }_3,{\beta }_4 or {\beta }_5$is non-zero.

$$\begin{gathered} \text{Test} \text{statistic}=\frac{\frac{\left(SS{E}_R-SS{E}_C\right)}{\left(k-g\right)}}{\frac{SS{E}_C}{\left[n-\left(k+1\right)\right]}} \\ =\frac{\frac{\left(3197.16-1830.44\right)}{\left(3-1\right)}}{\frac{1830.44}{\left[40-\left(3+1\right)\right]}} \\ =13.4401 \end{gathered}$$

For$\alpha =.05$,F-Test statistic = 13.4401.

The numerator degree of freedom is k – g =2, and the denominator degree of freedom is n – (k+1) = 36.

We know that it is rejected if

The critical value of F_0.05 = 3.37, since F-statistic>F_0.05, therefore, at95%confidence interval we reject H_0.

Therefore, at least one of the parameters${\mathit{\beta }}_{\mathbf{3}}\mathbf{,}{\mathit{\beta }}_{\mathbf{4}}\mathbf{ }\mathit{o}\mathit{r}\mathbf{ }{\mathit{\beta }}_{\mathbf{5}}$ is non-zero.

(e) Interpretation

The results of the hypothesis testing show that the complete model fits the data better. It would be more useful to predict y.