Chapter 16, Problem 133CE

a:

To determine

Calculate the coefficient.

Explanation of Solution

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

$\begin{array}{c}{\text{b}}_{\text{1}}=\frac{{\text{s}}_{\text{xy}}}{{\text{s}}_{\text{x}}^{\text{2}}}\\ =\frac{2538}{20,690}\\ =0.123\end{array}$

The value of coefficient is 0.123.

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

$\begin{array}{c}{\text{b}}_{\text{0}}=\overline{\text{y}}-{\text{b}}_{\text{1}}\overline{\text{x}}\\ = 318.6–0.123\left(300\right)\\ =281.7\end{array}$

The value of intercept is 281.7.

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:

$\widehat{\text{y}}=\text{281}\text{.7}+\text{0}\text{.123x}$

The coefficient (Slope) indicates that increasing one unit of independent variable (fertilizer) leads to an increase in the dependent variable (yield) by 0.123.

b:

To determine

Testing the hypothesis.

Explanation of Solution

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

$\begin{array}{c}\text{SSE}=\text{(n}-\text{1)}\left({\text{s}}_{\text{y}}^{\text{2}}-\frac{{\text{s}}_{\text{xy}}^{\text{2}}}{{\text{s}}_{\text{x}}^{\text{2}}}\right)\\ =(30-1)\left(5,230-\frac{{(2,538)}^2}{20,690}\right)\\ =142,641\end{array}$

Error sum of square is 142.641.

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

$\begin{array}{c}{\text{s}}_{\text{ε}}=\sqrt{\frac{\text{SSE}}{\text{n}-\text{2}}}\\ =\sqrt{\frac{142,641}{30-2}}\\ =71.37\end{array}$

The value of ${\text{s}}_{\text{ε}}$ is 71.37.

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:

$\begin{array}{l}\text{t}>{\text{t}}_{\frac{\text{α}}{2}\text{,n}-\text{2}}\\ ={\text{t}}_{\frac{\text{0}\text{.05}}{2}\text{,30}-2}\\ =\text{2}\text{.048}\end{array}$

The t table value is 2.048.

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

$\begin{array}{c}{\text{s}}_{{\text{b}}_{\text{1}}}=\frac{{\text{s}}_{\text{ε}}}{\sqrt{\text{(n}-{\text{1)s}}_{\text{x}}^{\text{2}}}}\\ =\frac{71.37}{\sqrt{(30-1)(20,690)}}\\ =0.0921\end{array}$

The value of ${\text{s}}_{{\text{b}}_{\text{1}}}$ is 0.092.

Calculated t-value can be obtained as follows:

$\begin{array}{c}\text{t}=\frac{{\text{b}}_{\text{1}}-{\text{β}}_{\text{1}}}{{\text{s}}_{{\text{b}}_{\text{1}}}}\\ =\frac{0.123-0}{0.0921}\\ =1.34\end{array}$

The t-value is 1.34. Since the calculated t-value is less than the t table value, the alternate hypothesis is not accepted.

c:

To determine

Coefficient of determination.

Explanation of Solution

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

$\begin{array}{c}{\text{R}}^{\text{2}}=\frac{{\text{s}}_{\text{xy}}^{\text{2}}}{{\text{s}}_{\text{x}}^{\text{2}}{\text{s}}_{\text{y}}^{\text{2}}}\\ =\frac{{(2,538)}^2}{(20,690)(5,230)}\\ =0.0595\end{array}$

The coefficient for the determination is 0.0851. The R² value indicates that 5.95% of the variation in the dependent variable is explained by the independent variable.

d:

To determine

Interpretation.

Explanation of Solution

Since the R² value is very low, it indicates that a weak relationship exists between he independent and dependent variables. Thus, the prediction would not be accurate.