Chapter 9.5, Problem 86E

(a)

To determine

To Find: The cubic model for the data.

Answer to Problem 86E

The required model is $f\left(t\right)=0.01417t^3-0.4515t^2+4.889t-6.27$ .

Explanation of Solution

Given:

The given data is shown in Table 1

Table 1

Year	Average Price $f\left(t\right)$
200	10.65
2008	11.26
2009	11.51
2010	11.54
2011	11.72
2012	11.88
2.13	12.13
2014	12.52

The variable $t=7$ corresponds to the year 2007.

Calculation:

Input the values on $L1$ and $f\left(t\right)$ values on $L2$ as shown in Figure 1

  

Figure 1

Use cubicreg to find the model as shown in Figure 2

  

Figure 2

Thus, the required model is,

  $f\left(t\right)=0.01417t^3-0.4515t^2+4.889t-6.27$

(b)

To determine

To Find: The plot for the data and the model in the same viewing window.

Answer to Problem 86E

The required graph is shown in Figure 1

Explanation of Solution

Consider the model is,

  $f\left(t\right)=0.01417t^3-0.4515t^2+4.889t-6.27$

From the above model and the give data the required plot is shown in Figure 3

  

Figure 3

(c)

To determine

To Find: The binomial coefficient to write $g\left(t\right)=f\left(t+5\right)$ in the standard form.

Answer to Problem 86E

The required expression is $g\left(t+5\right)=0.1417t^3-0.23895t^2+1.43675t+8.65875$ .

Explanation of Solution

Consider the model for the data set is,

  $f\left(t\right)=0.01417t^3-0.4515t^2+4.889t-6.27$

Then, the value of $g\left(t\right)=f\left(t+5\right)$ is evaluated as,

  $\begin{array}{c}g\left(t+5\right)=0.01417{\left(t+5\right)}^3-0.4515{\left(t+5\right)}^2+4.889\left(t+5\right)-6.27\\ =0.1417t^3-0.23895t^2+1.43675t+8.65875\end{array}$

(d)

To determine

To find: the graph for $g$ in the same viewing.

Answer to Problem 86E

The required graph is shown in Figure 4

Explanation of Solution

Consider the model for the data set is,

  $\begin{array}{l}f\left(t\right)=0.01417t^3-0.4515t^2+4.889t-6.27\\ g\left(t+5\right)=0.1417t^3-0.23895t^2+1.43675t+8.65875\end{array}$

Then, the graph for $g\left(t\right)$ and $f\left(t\right)$ is shown in Figure 4

  

Figure 4

(e)

To determine

To find: The average price in 2015.

Answer to Problem 86E

The increase in average price is same for both the models that is $13.3\text{cents}/\text{kWh}$ .

Explanation of Solution

Consider the model for the data set is,

  $\begin{array}{l}f\left(t\right)=0.01417t^3-0.4515t^2+4.889t-6.27\\ g\left(t+5\right)=0.1417t^3-0.23895t^2+1.43675t+8.65875\end{array}$

Then, the average price in 2015 is obtained by substituting $t=15$ in the model as,

  $\begin{array}{c}f\left(15\right)=0.01417{\left(15\right)}^3-0.4515{\left(15\right)}^2+4.889\left(15\right)-6.27\\ =13.3\text{cents}/\text{kWh}\end{array}$

Also,

  $\begin{array}{c}g\left(5+5\right)=0.1417{\left(5+5\right)}^3-0.23895{\left(5+5\right)}^2+1.43675\left(5+5\right)+8.65875\\ =13.3\text{cents}/\text{kWh}\end{array}$

(f)

To determine

To find: The answer of part (e) is reasonable.

Answer to Problem 86E

The answer of part (e) is not reasonable.

Explanation of Solution

Since, in part (e) the increase in reasonable price obtained is much more than the increase in past 7 years therefore the answer of part (e) is not reasonable.

(g)

To determine

To find: The factor that contribute to the change in the average price.

Answer to Problem 86E

The affecting factors are fuel cost, operating cost of power plants, transmission and distribution system, weather conditions, and regulations.

Explanation of Solution

There are several factors like fuel cost, operating cost of power plants, transmission and distribution system, weather conditions, and regulations.