Chapter 16, Problem 131CE

a:

To determine

Calculate the coefficient.

Explanation of Solution

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

$\begin{array}{c}{\text{b}}_{\text{1}}=\frac{{\text{s}}_{\text{xy}}}{{\text{s}}_{\text{x}}^{\text{2}}}\\ =\frac{74.02}{3.47}\\ =21.33\end{array}$

The value of coefficient is 21.33.

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

$\begin{array}{c}{\text{b}}_{\text{0}}=\overline{\text{y}}-{\text{b}}_{\text{1}}\overline{\text{x}}\\ = 384.81–21.33\left(4.12\right)\\ =296.93\end{array}$

The value of intercept is 296.93.

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:

$\widehat{\text{y}}=\text{296}\text{.93}+\text{21}\text{.33x}$

b:

To determine

Interpretation.

Explanation of Solution

The coefficient indicates that increasing one unit of independent variable (advertisement) leads to increase in the dependent variable (customer) by 21.33.

c:

To determine

Testing the hypothesis.

Explanation of Solution

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

$\begin{array}{c}\text{SSE}=\text{(n}-\text{1)}\left({\text{s}}_{\text{y}}^{\text{2}}-\frac{{\text{s}}_{\text{xy}}^{\text{2}}}{{\text{s}}_{\text{x}}^{\text{2}}}\right)\\ =(26-1)\left(18,552-\frac{{(74.02)}^2}{3.47}\right)\\ =424,326\end{array}$

Error sum of square is 424,326.

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

$\begin{array}{c}{\text{s}}_{\text{ε}}=\sqrt{\frac{\text{SSE}}{\text{n}-\text{2}}}\\ =\sqrt{\frac{424,326}{26-2}}\\ =132.97\end{array}$

The value of ${\text{s}}_{\text{ε}}$ is 132.97.

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

The t table value can be calculated as follows:

$\begin{array}{l}\text{t}>{\text{t}}_{\text{α,n}-\text{2}}\\ ={\text{t}}_{\text{0}\text{.05,50}-2}\\ =\text{1}\text{.711}\end{array}$

The t table value is 1.711.

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

$\begin{array}{c}{\text{s}}_{{\text{b}}_{\text{1}}}=\frac{{\text{s}}_{\text{ε}}}{\sqrt{\text{(n}-{\text{1)s}}_{\text{x}}^{\text{2}}}}\\ =\frac{132.97}{\sqrt{(26-1)(3.47)}}\\ =14.28\end{array}$

The value of ${\text{s}}_{{\text{b}}_{\text{1}}}$ is 14.28.

Calculated t-value can be obtained as follows:

$\begin{array}{c}\text{t}=\frac{{\text{b}}_{\text{1}}-{\text{β}}_{\text{1}}}{{\text{s}}_{{\text{b}}_{\text{1}}}}\\ =\frac{21.33-0}{14.28}\\ =1.49\end{array}$

The t-value is 1.49. Since the calculated t-value is less than the t table value, the alternate hypothesis is not accepted.

d:

To determine

Coefficient of determination.

Explanation of Solution

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

$\begin{array}{c}{\text{R}}^{\text{2}}=\frac{{\text{s}}_{\text{xy}}^{\text{2}}}{{\text{s}}_{\text{x}}^{\text{2}}{\text{s}}_{\text{y}}^{\text{2}}}\\ =\frac{{(74.02)}^2}{(3.47)(18,552)}\\ =0.0851\end{array}$

The coefficient for the determination is 0.0851. The R² value indicates that a weak linear relationship exists between the independent variable and dependent variable.

e:

To determine

Interpretation.

Explanation of Solution

Since the model reveals that a weak relationship exists between the independent and dependent variables, the prediction would not be accurate.