Chapter 23, Problem 40E

(a)

To determine

To discuss the assumptions and conditions for using Student’s t inference methods with these data.

Explanation of Solution

In the question, they are studying about should we generate electricity with our own personal wind turbine or not. That depends on whether you have enough wind on your site. Thus, a candidate site was monitored for a year with wind speeds recorded every six hours. Thus, we have,

  $\begin{array}{l}n=1114\\ \mu =8.019\text{mph}\\ \sigma =3.813\text{mph}\end{array}$

Moreover, the histogram between the wind speed and the recordings are given in the question along with time plot and the scatterplot of the following data recorded. Thus, the assumptions and conditions for using Student’s t inference methods with these data are as,

Random condition: It is satisfied assuming that the winds speed recorded on a site were randomly selected.

Independent condition: It is satisfied since the sample of $1114$ wind speed recorded on a site is less than $10\%$ of the population of all wind speed recorded on a site.

Normal condition: It is satisfied because by looking at the histogram of the recorded winds and the winds speed we can say that it is approximately normal in shape.

Thus, all the conditions and assumptions are satisfied.

(b)

To determine

To explain what would you tell the landowner about whether this is suitable for a small wind turbine.

Answer to Problem 40E

Yes, this is suitable for a small wind turbine.

Explanation of Solution

In the question, they are studying about should we generate electricity with our own personal wind turbine or not. That depends on whether you have enough wind on your site. Thus, a candidate site was monitored for a year with wind speeds recorded every six hours. Thus, we have,

  $\begin{array}{l}n=1114\\ \mu =8.019\text{mph}\\ \sigma =3.813\text{mph}\end{array}$

Moreover, the histogram between the wind speed and the recordings are given in the question along with time plot and the scatterplot of the following data recorded. Thus, for the claim we need to test. Let us define the hypothesis for testing:

  $\begin{array}{l}{H}_0:\mu =8\\ H_1:\mu >8\end{array}$

Thus the value of the test statistics will be:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{8.019-8}{\frac{3.813}{\sqrt{1114}}}\\ =0.166\end{array}$

The degrees of freedom will be as according to the sample given:

  $\begin{array}{l}df=n-1\\ =1114-1\\ =1113\end{array}$

Now the P-value according to the degrees of freedom will be as:

  $P<0.17$

As we know that if the P-value is smaller than the significance level, then the null hypothesis is rejected. Thus,

  $P>0.05=5\%\Rightarrow \text{Fail to Reject }{H}_0$

Thus, we conclude that the landowners could use small wind turbines for the site.