Chapter 14, Problem 8E

The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars.

x₁ is the number of architects employed by the company.

x₂ is the number of engineers employed by the company.

x₃ is the number of years involved with health care projects.

x₄ is the number of states in which the firm operates.

x₅ is the percent of the firm’s work that is health care–related.

1. a. Write out the regression equation.
2. b. How large is the sample? How many independent variables are there?
3. c. Conduct a global test of hypothesis to see if any of the set of regression coefficients could be different from 0. Use the .05 significance level. What is your conclusion?
4. d. Conduct a test of hypothesis for each independent variable. Use the .05 significance level. Which variable would you consider eliminating first?
5. e. Outline a strategy for deleting independent variables in this case.

To determine

Provide the regression equation.

Answer to Problem 8E

The multiple regression equation is,

$\widehat{y}=7.987+0.122x_1-1.220x_2-0.063x_3+0.523x_4-0.065x_5$.

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 problem, y defines the total amount of fees, $x_1$ defines the number of architects by the company, $x_2$ defines the total number of engineers employed by the company, $x_3$ defines the number of years involved with health projects, $x_4$ is the number of states in which the film operates and $x_5$ is the percent of the firm’s work, which is related with health care.

In the given problem, there are five independent variables $x_1,x_2,x_3,x_4\text{and}{x}_5$ with coefficients 0.122, –1.220, –0.063, 0.523 and –0.065, respectively and the intercept term is 7.987.  Consider that the dependent variable is y.

Hence, the multiple regression equation is,

$\widehat{y}=7.987+0.122x_1-1.220x_2-0.063x_3+0.523x_4-0.065x_5$.

To determine

Find the sample size and the number of independent variables.

Answer to Problem 8E

The sample size and the number of independent variables are 65 and 2, respectively.

Explanation of Solution

Calculation:

In an ANOVA table the total degrees of freedom is defined as $df=n-1$, where n is the sample size.

According to the given question the total degrees of freedom is 51.

That is,

$\begin{array}{c}51=n-1\\ n=51+1\\ =52\end{array}$

Therefore, the sample size is 52.

In an ANOVA table the degrees of freedom of regression is defined as k, where k is the number of independent variables.

According to the given question, the degrees of freedom for regression is 5. That is, $k=5$.

Hence, the number of independent variables is 5.

To determine

Perform a global hypothesis test to check whether any of the set of regression coefficients are different from 0 at 0.05 significance level.

Provide the conclusion.

Answer to Problem 8E

There is strong evidence that not all the regression coefficients are equal to 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 $i=1,2,3,4,5$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_1={\beta }_2={\beta }_3={\beta }_4={\beta }_5=0$.

That is, the model is not significant.

Alternative hypothesis:

$H_1:\text{Not all }{\beta }_i\text{'}\text{s are equal to }0$.

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 given ANOVA table the value of F test statistic is 12.89 with numerator degrees of freedom of 5 and denominator degrees of freedom 46.

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 not all the regression coefficients are equal to 0 at 0.05 significance level. Thus, there are one or more variables for which the model is not significant.

To determine

Perform an individual test of each independent variable at 0.05 significance level.

Find the variable that can be eliminated.

Answer to Problem 8E

There is significant relation between y and $x_1,x_2\text{and}{x}_4$, whereas there is no significant relation between y and $x_3\text{and}{x}_5$.

The independent variables $x_3\text{and}{x}_5$ can be eliminated.

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 given ANOVA table the value of t test statistic corresponding to $x_1$ is 3.9209 with 46 degrees of freedom.

The level of significance is $\alpha =0.05$.

Conclusion:

Here, p-value corresponding to the $x_1$ 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 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 given ANOVA table the value of t test statistic corresponding to $x_2$ is –2.270 with 46 degrees of freedom.

Conclusion:

Here, p-value corresponding to the $x_2$ is 0.028.

Hence, $p\text{-value}\left(=0.021\right)<\alpha \left(=0.05\right)$.

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 $x_2$.

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 given ANOVA table the value of t test statistic corresponding to $x_3$ is –1.610 with 46 degrees of freedom.

Conclusion:

Here, p-value corresponding to the $x_3$ is 0.114.

Hence, $p\text{-value}\left(=0.114\right)>\alpha \left(=0.05\right)$.

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 $x_3$.

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 given ANOVA table the value of t test statistic corresponding to $x_4$ is 3.6900 with 46 degrees of freedom.

Conclusion:

Here, p-value corresponding to the $x_4$ is 0.001.

Hence, $p\text{-value}\left(=0.001\right)<\alpha \left(=0.05\right)$.

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 $x_4$.

For independent variable $x_3$:

Consider that ${\beta }_5$ is the population regression coefficient of independent variable $x_5$.

State the hypotheses:

Null hypothesis:

$H_0:{\beta }_5=0$.

That is, there is no significant relationship between y and $x_5$.

Alternative hypothesis:

$H_1:{\beta }_5\ne 0$.

That is, there is significant relationship between y and $x_5$.

According to the given ANOVA table the value of t test statistic corresponding to $x_5$ is –1.620 with 46 degrees of freedom.

Conclusion:

Here, p-value corresponding to the $x_5$ is 0.112.

Hence, $p\text{-value}\left(=0.112\right)>\alpha \left(=0.05\right)$.

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 $x_5$.

As there are no significant relationship between the dependent variable and the independent variables $x_3\text{and}{x}_5$, it is better to eliminate these two variables.

To determine

Provide a plan for possibly removing independent variables.

Explanation of Solution

From Part (e), it is found that there are no significant relationship between the dependent variable and the independent variables $x_3\text{and}{x}_5$. Hence, it is better to eliminate these two variables $x_3\text{and}{x}_5$; and perform the regression analysis only with $x_1,x_2\text{and}{x}_4$ as independent variable.

Thus, it can be said that the total amount of fees can be predicted well if it can be estimated using only the number of architects, the number of engineers in the company and the number of stated in which the film operate.