Chapter 11, Problem 8E

(a)

To determine

To test: The hypothesis that the coefficient of variable $x_1$ is zero against the hypothesis that it is not zero.

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable $x_1$ is zero is true.

Explanation of Solution

Given: The confidence level is provided as $95\%$. The following information has been provided in the problem:

$\begin{array}{c}\widehat{y}=1.6+6.4x_1+5.7x_2\\ n=25\\ {\text{SE}}_{b_1}=3.3\end{array}$

Explanation:

Calculation: In a regression equation, the coefficient of $x_1$ is denoted as ${\beta }_1$. According to the provided problem, the null hypothesis will be formulated as follows:

$H_0:{\beta }_1=0$

The alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

$H_a:{\beta }_1\ne 0$

The total number of observations has been provided in the problem as

$n=25$

Since, there are 2 explanatory variables in the regression equation and the degrees of freedom for the provided problem can be calculated as follows:

$\begin{array}{c}n-p-1=25-2-1\\ =22\end{array}$

The value of coefficient of $x_1$ is $b_1=6.4$. Since, the test is two-tailed, the value of $t^{*}$ for $95\%$ confidence level for 22 degrees of freedom is obtained from the table as: 2.074. The $t\text{-statistic}$ is calculated as:

$\begin{array}{c}t=\frac{b_1}{{\text{SE}}_{b_1}}\\ =\frac{6.4}{3.3}\\ =1.94\end{array}$

Conclusion: Since, the calculated value of $t$ is less than the observed value of $t$, the null hypothesis does not get rejected. Hence, the null hypothesis that the coefficient of variable $x_1$ is zero is true.

(b)

To determine

To test: The hypothesis that the coefficient of variable $x_1$ is zero against the hypothesis that it is not zero.

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable $x_1$ is zero is not true.

Explanation of Solution

Given: The confidence level is provided as $95\%$. The following information has been provided in the problem:

$\begin{array}{c}\widehat{y}=1.6+6.4x_1+5.7x_2\\ n=43\\ {\text{SE}}_{b_1}=2.9\end{array}$

Explanation:

Calculation: In a regression equation, the coefficient of $x_1$ is denoted as ${\beta }_1$. According to the provided problem, the null hypothesis will be formulated as follows:

$H_0:{\beta }_1=0$

And the alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

$H_a:{\beta }_1\ne 0$

The total number of observations has been provided in the problem as:

$n=43$

Since, there are 2 explanatory variables in the regression equation, the degrees of freedom for the provided problem can be calculated as follows:

$\begin{array}{c}n-p-1=43-2-1\\ =40\end{array}$

The value of coefficient of $x_1$ is $b_1=6.4$. Since, the test is two-tailed, the value of $t^{*}$ for $95\%$ confidence level for 40 degrees of freedom is obtained from the table as: 2.021. $t\text{-statistic}$ is calculated as:

$\begin{array}{c}t=\frac{b_1}{{\text{SE}}_{b_1}}\\ =\frac{6.4}{2.9}\\ =2.206\end{array}$

Conclusion: Since, the calculated value of $t$ is more than the observed value of $t$, the null hypothesis gets rejected. Hence, the null hypothesis that the coefficient of variable $x_1$ is zero is not true.

(c)

To determine

To test: The hypothesis that the coefficient of variable $x_1$ is zero against the hypothesis that it is not zero.

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable $x_1$ is zero is true.

Explanation of Solution

Given: The confidence level is provided as $95\%$. The following information has been provided in the problem:

$\begin{array}{c}\widehat{y}=1.6+4.8x_1+3.2x_2+5.2x_3\\ n=25\\ {\text{SE}}_{b_1}=2.7\end{array}$

Explanation:

Calculation: In a regression equation, the coefficient of $x_1$ is denoted as ${\beta }_1$. According to the provided problem, the null hypothesis will be formulated as follows:

$H_0:{\beta }_1=0$

And the alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

$H_a:{\beta }_1\ne 0$

The total number of observations has been provided in the problem as:

$n=25$

Since, there are 3 explanatory variables in the regression equation, the degrees of freedom for the provided problem can be calculated as follows:

$\begin{array}{c}n-p-1=25-3-1\\ =21\end{array}$

The value of coefficient of $x_1$ is $b_1=4.8$. Since, the test is two-tailed, the value of $t^{*}$ for $95\%$ confidence level for 21 degrees of freedom is obtained from the table as: 2.080. The $t\text{-statistic}$ is calculated as:

$\begin{array}{c}t=\frac{b_1}{{\text{SE}}_{b_1}}\\ =\frac{4.8}{2.7}\\ =1.777\end{array}$

Conclusion: Since, the calculated value of $t$ is less than the observed value of $t$, the null hypothesis does not get rejected. Hence, the null hypothesis that the coefficient of variable $x_1$ is zero is true.

(d)

To determine

To test: The hypothesis that the coefficient of variable $x_1$ is zero against the hypothesis that it is not zero.

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable $x_1$ is zero is not true.

Explanation of Solution

Given: The confidence level is provided as $95\%$. The following information has been provided in the problem:

$\begin{array}{c}\widehat{y}=1.6+4.8x_1+3.2x_2+5.2x_3\\ n=104\\ {\text{SE}}_{b_1}=1.8\end{array}$

Explanation:

Calculation: In a regression equation, the coefficient of $x_1$ is denoted as ${\beta }_1$. According to the provided problem, the null hypothesis will be formulated as follows:

$H_0:{\beta }_1=0$

And the alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

$H_a:{\beta }_1\ne 0$

The total number of observations has been provided in the problem as:

$n=104$

Since, there are 3 explanatory variables in the regression equation, the degrees of freedom for the provided problem can be calculated as follows:

$\begin{array}{c}n-p-1=104-3-1\\ =100\end{array}$

The value of coefficient of $x_1$ is $b_1=4.8$. Since, the test is two-tailed, the value of $t^{*}$ for $95\%$ confidence level for 100 degrees of freedom is obtained from the table as: 1.984. The $t\text{-statistic}$ is calculated as:

$\begin{array}{c}t=\frac{b_1}{{\text{SE}}_{b_1}}\\ =\frac{4.8}{1.8}\\ =2.67\end{array}$

Conclusion: Since, the calculated value of $t$ is more than the observed value of $t$, the null hypothesis gets rejected. Hence, the null hypothesis that the coefficient of variable $x_1$ is zero is not true.