Chapter 16, Problem 132CE

a:

To determine

Calculate the coefficient.

Explanation of Solution

The value of coefficient (b₁) can be calculated as follows:

$\begin{array}{c}{\text{b}}_{\text{1}}=\frac{{\text{s}}_{\text{xy}}}{{\text{s}}_{\text{x}}^{\text{2}}}\\ =\frac{936.82}{378.77}\\ =2.47\end{array}$

The value of coefficient is 2.47.

The intercept (b₀) value can be calculated as follows:

$\begin{array}{c}{\text{b}}_{\text{0}}=\overline{\text{y}}-{\text{b}}_{\text{1}}\overline{\text{x}}\\ =395.21–2.47\left(113.35\right)\\ =115.24\end{array}$

The value of intercept is 115.24.

The general regression equation can be written as follows:

$\widehat{\text{y}}={\text{b}}_{\text{0}}+{\text{b}}_{\text{1}}\text{x}$

The terms $\widehat{\text{y}}$ is dependent variable (Estimated value of dependent value), the term b₀ is indicates intercept, the term x indicates independent variable, and the term b₁ indicates the coefficient of independent variable.

Thus, the sample regression equation can be written as follows:

$\widehat{\text{y}}=\text{114}\text{.85}+\text{2}\text{.47x}$

b:

To determine

Interpretation.

Explanation of Solution

The coefficient indicates that increasing one unit of independent variable (month of age) leads to increase in the dependent variable (repair cost) by 2.47. The value of intercept (b₀) is 115.24.

c:

To determine

Coefficient of determination.

Explanation of Solution

Coefficient of determination i $\left({\text{R}}^{\text{2}}\right)$ can be calculated as follows:

$\begin{array}{c}{\text{R}}^{\text{2}}=\frac{{\text{s}}_{\text{xy}}^{\text{2}}}{{\text{s}}_{\text{x}}^{\text{2}}{\text{s}}_{\text{y}}^{\text{2}}}\\ =\frac{{(936.82)}^2}{(378.77)(4,094.79)}\\ =0.5659\end{array}$

The coefficient for the determination is 0.5659. The R² value indicates that 56.59% of the variation in the dependent variable is explained by the independent variable.

d:

To determine

Testing the hypothesis.

Explanation of Solution

Error sum of square (SSE) can be calculated as follows:

$\begin{array}{c}\text{SSE}=\text{(n}-\text{1)}\left({\text{s}}_{\text{y}}^{\text{2}}-\frac{{\text{s}}_{\text{xy}}^{\text{2}}}{{\text{s}}_{\text{x}}^{\text{2}}}\right)\\ =(20-1)\left(4,094.79-\frac{{(936.82)}^2}{378.77}\right)\\ =33,777\end{array}$

Error sum of square is 33,777.

The value of ${\text{s}}_{\text{ε}}$ can be calculated as follows:

$\begin{array}{c}{\text{s}}_{\text{ε}}=\sqrt{\frac{\text{SSE}}{\text{n}-\text{2}}}\\ =\sqrt{\frac{33,777}{20-2}}\\ =43.32\end{array}$

The value of ${\text{s}}_{\text{ε}}$ is 43.32.

Null hypothesis (H₀) is ${\text{β}}_{\text{1}}=0$ and alternate hypothesis (H₁) ${\text{β}}_{\text{1}}\ne 0$.

The t table value can be calculated as follows:

$\begin{array}{l}\text{t}>{\text{t}}_{\frac{\text{α}}{2}\text{,n}-\text{2}}\\ ={\text{t}}_{\frac{\text{0}\text{.05}}{2}\text{,20}-2}\\ =\text{2}\text{.101}\end{array}$

The t table value is 2.101.

The value of ${\text{s}}_{{\text{b}}_{\text{1}}}$ can be calculated as follows:

$\begin{array}{c}{\text{s}}_{{\text{b}}_{\text{1}}}=\frac{{\text{s}}_{\text{ε}}}{\sqrt{\text{(n}-{\text{1)s}}_{\text{x}}^{\text{2}}}}\\ =\frac{43.32}{\sqrt{(20-1)(378.77)}}\\ =0.511\end{array}$

The value of ${\text{s}}_{{\text{b}}_{\text{1}}}$ is 0.511.

Calculated t value can be obtained as follows:

$\begin{array}{c}\text{t}=\frac{{\text{b}}_{\text{1}}-{\text{β}}_{\text{1}}}{{\text{s}}_{{\text{b}}_{\text{1}}}}\\ =\frac{2.47-0}{.511}=4.84\end{array}$

The t-value is 4.84. Since the calculated t-value is greater than the t table value, the alternate hypothesis is accepted.

e:

To determine

Calculate the lower and the upper class limit.

Explanation of Solution

The estimated value of y $\left(\widehat{\text{y}}\right)$ can be calculated as follows.

$\begin{array}{c}\widehat{\text{y}}={\text{b}}_{\text{0}}+{\text{b}}_{\text{1}}{\text{x}}_{\text{g}}\\ =115.24+2.47(120)\\ =411.64\end{array}$

The estimated value of y is 411.64.

Prediction interval (PI) of sales can be calculated as follows:

$\begin{array}{c}\text{PI}=\widehat{\text{y}}\pm {\text{t}}_{\frac{\text{α}}{\text{2}}\text{,n}-\text{2}}{\text{×s}}_{\text{ε}}\sqrt{\text{1}+\frac{\text{1}}{\text{n}}+\frac{{{\text{(x}}_{\text{g}}-\overline{\text{x}}\text{)}}^{\text{2}}}{\text{(n}-{\text{1)s}}_{\text{x}}^{\text{2}}}}\\ =\widehat{\text{y}}\pm {\text{t}}_{\frac{\text{0}\text{.05}}{\text{2}}\text{,20}-\text{2}}{\text{×s}}_{\text{ε}}\sqrt{\text{1}+\frac{\text{1}}{\text{n}}+\frac{{{\text{(x}}_{\text{g}}-\overline{\text{x}}\text{)}}^{\text{2}}}{\text{(n}-{\text{1)s}}_{\text{x}}^{\text{2}}}}\\ =411.64\pm 2.101(43.32)\sqrt{1+\frac{1}{20}+\frac{{(120-113.35)}^2}{(20-1)(378.77)}}\\ =318.1\text{ and 505}\text{.2}\end{array}$

The lower class limit is 318.1 and the upper class limit is 505.2.