Chapter 14, Problem 25CE

a.

To determine

Find the regression equation.

Answer to Problem 25CE

The regression equation is $\underset{\_}{\widehat{y}=652+13.42x_1-6.71x_2+205.6x_3-33.5x_4}$.

Explanation of Solution

Calculation:

Multiple linear regression model:

A multiple linear regression model is given as $\widehat{y}=a+b_1{x}_1+b_2{x}_2+b_3{x}_3+\mathrm{...}+b_k{x}_k$ where y is the response or dependent variable, and $x_1,x_2,\mathrm{...},x_k$ are the k quantitative independent variables where k is a positive integer.

Here, a is the intercept term of the regression model, that is, the value of predicted value of y when X’s are 0 and $b_i$’s are the slopes, that is, the amount of change of the predicted value of y for one unit increase in $x_i$ when all other independent variables are constant.

In the given part the predicted dependent variable y is the monthly salary and the length of service $\left(x_1\right)$, the age $\left(x_2\right)$, the gender $\left(x_3\right)$ and the job $\left(x_4\right)$ are the independent variables.

Step by step procedure to obtain the regression equation using MINITAB software:

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of Salary.
• Under Continuous predictors, enter the columns of Service, Age, Gender and Job.
• Click OK.

Output using MINITAB software is given below:

Thus, the regression equation is $\underset{\_}{\widehat{y}=652+13.42x_1-6.71x_2+205.6x_3-33.5x_4}$.

b.

To determine

Find the value of $R^2$.

Also explain about the $R^2$.

Answer to Problem 25CE

The value of $R^2$ is 43.27%.

Explanation of Solution

According to output in Part (a), the value of $R^2$ is 43.27%.

Hence, it can be said that only 43.27% variability in the monthly salary is explained by the length of service, age, gender and the job designation of the employees, using the regression equation.

c.

To determine

Perform a global test to find whether any of the independent variables are different from 0.

Answer to Problem 25CE

There is enough evidence that any of the independent variables are different from 0 at 0.05 significance level.

Explanation of Solution

Calculation:

Consider that y is dependent variable and $x_i\text{'}\text{s}$ are the independent variables where ${\beta }_i\text{'}\text{s}$ are the corresponding population regression coefficient for all $i=1,2,3,4$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_1={\beta }_2={\beta }_3={\beta }_4=0$.

That is, the model is not significant.

Alternative hypothesis:

$H_1:\text{At least one }{\beta }_i\text{ is not equal to }0$.

That is, the model is significant.

In case of global test the F test statistic is defined as,

$F=\frac{\frac{\text{SSR}}{k}}{\frac{\text{SSE}}{n-k-1}}$, where SSR, SSE, n and k are the regression sum of square, error sum of square, sample size and the number of independent variables.

According to the output from Part (a) the value of F statistic is 4.77 with numerator degrees of freedom 4 and denominator degrees of freedom 25.

Consider that, the level of significance is $\alpha =0.05$.

Decision rule:

• If $p\text{-value}\le \alpha$, then reject the null hypothesis.
• Otherwise failed to reject the null hypothesis.

Conclusion:

Here, p-value corresponding to the global test is 0.

Hence, $p\text{-value}\left(=0\right)<\alpha \left(=0.05\right)$.

That is, the p-value is less than the level of significance.

Therefore, reject the null hypothesis.

Hence, it can be concluded that any of the independent variables are different from 0 at 0.05 significance level.

d.

To determine

Perform an individual test to determine whether any of the independent variables can be dropped.

Answer to Problem 25CE

There is no significant relationship between the dependent variable “Salary” and the independent variables “Age” and “Job”, thus it is better to omit these two variables.

Explanation of Solution

Calculation:

For independent variable $x_1$:

Consider that ${\beta }_1$ is the population regression coefficient of independent variable $x_1$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_1=0$.

That is, there is no significant relationship between y and $x_1$.

Alternative hypothesis:

$H_1:{\beta }_1\ne 0$.

That is, there is significant relationship between y and $x_1$.

In case of individual regression coefficient test the t test statistic is defined as,

$t=\frac{b_i}{s_{b_i}}$, where $b_i$ and $s_{b_i}$ are the i^th regression coefficient and the standard deviation of the i^th regression coefficient.

According to the output in Part (a) the t statistic value corresponding to $x_1$ is 2.61 with 25 degrees of freedom.

Conclusion:

Here, p-value corresponding to the “Service”$\left(x_1\right)$ is 0.015.

Hence, $p\text{-value}\left(=0\right)<\alpha \left(=0.05\right)$.

That is, the p-value is less than the level of significance.

Therefore, reject the null hypothesis.

Hence, it can be concluded that there is significant relationship between y and $x_1$.

For independent variable $x_2$:

Consider that ${\beta }_2$ is the population regression coefficient of independent variable $x_2$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_2=0$.

That is, there is no significant relationship between y and $x_2$.

Alternative hypothesis:

$H_1:{\beta }_2\ne 0$.

That is, there is significant relationship between y and $x_2$.

According to the output in Part (a) the value of t test statistic corresponding to $x_2$ is –1.06 with 25 degrees of freedom.

Conclusion:

Here, p-value corresponding to the “Age”$\left(x_2\right)$ is 0.301.

Hence, $p\text{-value}\left(=0.301\right)>\alpha \left(=0.05\right)$.

That is, the p-value is greater than the level of significance.

Therefore, fail to reject the null hypothesis.

Hence, it can be concluded that there is no significant relationship between y and $x_2$.

For independent variable $x_3$:

Consider that ${\beta }_3$ is the population regression coefficient of independent variable $x_3$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_3=0$.

That is, there is no significant relationship between y and $x_3$.

Alternative hypothesis:

$H_1:{\beta }_3\ne 0$.

That is, there is significant relationship between y and $x_3$.

According to the output in Part (d) the value of t test statistic corresponding to $x_3$ is 2.28 with 25 degrees of freedom.

Conclusion:

Here, p-value corresponding to the “Gender”$\left(x_3\right)$ is 0.032.

Hence, $p\text{-value}\left(=0.032\right)<\alpha \left(=0.05\right)$.

That is, the p-value is less than the level of significance.

Therefore, reject the null hypothesis.

Hence, it can be concluded that there is significant relationship between y and $x_3$.

For independent variable $x_4$:

Consider that ${\beta }_4$ is the population regression coefficient of independent variable $x_4$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_4=0$.

That is, there is no significant relationship between y and $x_4$.

Alternative hypothesis:

$H_1:{\beta }_4\ne 0$.

That is, there is significant relationship between y and $x_4$.

According to the output in Part (a) the value of t test statistic corresponding to $x_4$ is –0.37 with 25 degrees of freedom.

Conclusion:

Here, p-value corresponding to the “Job”$\left(x_4\right)$ is 0.712.

Hence, $p\text{-value}\left(=0.712\right)>\alpha \left(=0.05\right)$.

That is, the p-value is greater than the level of significance.

Therefore, fail to reject the null hypothesis.

Hence, it can be concluded that there is no significant relationship between y and $x_4$.

As there is no significant relationship between the dependent variable “Salary” and the independent variables “Age” and “Job”, thus it is better to omit these two variables and perform the regression analysis only with the independent variables “Service” and “Gender”.

e.

To determine

Find the regression equation using the significant independent variables.

Find the amount of money does a man earn per month than a woman.

Explain whether there is any difference

Answer to Problem 25CE

The regression equation is $\underset{\_}{\widehat{y}=784+9.02x_1+224.4x_2}$.

Explanation of Solution

Calculation:

Dummy variable:

A dichotomous variable is defined as a dummy variable, where one outcome is defined as 1 another as 0.

In the Part the predicted dependent variable y is the monthly salary and the length of service $\left(x_1\right)$, and the gender $\left(x_2\right)$ are the independent variables.

The independent random variable $x_2$ is defined as dummy variable.

Hence,

$x_2=\{\begin{array}{l}1,\text{male}\\ \text{0,}\text{female}\end{array}$.

Step by step procedure to obtain the regression equation using MINITAB software:

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of Salary.
• Under Continuous predictors, enter the columns of Service, and Gender.
• Click OK.

Output using MINITAB software is given below:

Thus, the regression equation is $\underset{\_}{\widehat{y}=784+9.02x_1+224.4x_2}$.

Here, the coefficient of $x_2$ is 224.4. For $x_2=1\left(\text{Male}\right)$ the slope coefficient value is 224.4 and for $x_2=0\left(\text{Female}\right)$ the slope coefficient value is 0. Hence, it can be said that a man earns $224.4 more than a woman.

As there is no significant relationship between the monthly salary and the designation of the employees, hence whether the employee has a management or engineering position does not make any difference.