Chapter PVII, Problem 15RE

(a)

To determine

To create a $90\%$ confidence interval for the mean annual rainfall in LA.

Answer to Problem 15RE

We are $90\%$ confident that the average annual rainfall in LA is between $11.65\text{ and }17.39$ inches.

Explanation of Solution

In the question, the data of the rainfall in inches over the years in the table is given. Let us assume all the conditions and assumptions are met. Thus, to test the hypothesis for mean we will use the calculator $\text{TI}-89$ . In the STAT Tests menu, choose $2:$ T -Test. You must specify whether you are using data stored in a list or whether you will enter the mean, standard deviation, and size of your sample. You must also specify the hypothesized model mean and whether the test is to be two-tail, lower-tail, or upper-tail. Select whether the test is to be simply computed or whether to display the distribution curve and highlight the area corresponding to the P-value of the test. So, as $c=90\%$ then by using the calculator $\text{TI}-89$ , the confidence interval will be:

  $(11.65,17.39)$ .

So, we are $90\%$ confident that the average annual rainfall in LA is between $11.65\text{ and }17.39$ inches.

(b)

To determine

To find out how many years’ data would you need.

Answer to Problem 15RE

We need $46$ years’ data.

Explanation of Solution

In the question, the data of the rainfall in inches over the years in the table is given. Let us assume all the conditions and assumptions are met. Thus, it is given if we wanted to estimate the mean actual rainfall with a margin of error of only two inches, it is calculated as:

  $\begin{array}{l}ME=t\times \frac{\sigma }{\sqrt{n}}\\ \Rightarrow 2=1.72\times \frac{7.82}{\sqrt{n}}\\ \Rightarrow \frac{2}{1.72}=\frac{7.82}{\sqrt{n}}\\ \Rightarrow n=45.2\\ \Rightarrow n=46\text{ years}\end{array}$

(c)

To determine

To explain do these data suggest any change in annual rainfall as time passes and check for an association between rainfall and year.

Answer to Problem 15RE

These data do not suggest any change in annual rainfall as time passes.

Explanation of Solution

In the question, the data of the rainfall in inches over the years in the table is given. Let us assume all the conditions and assumptions are met. Let us define the hypothesis by:

Null hypothesis: There is no association between rainfall and years.

Alternative hypothesis: There is an association between rainfall and years.

Now, we will perform the regression analysis by using Excel. We will go to the data tab and then we will go to the data analysis option and then the dialogue box will appear and then we choose regression analysis from that. Thus, the result will be as follows:

Regression Analysis
	r²	0.001	n	22
	r	0.028	k	1
	Std. Error	8.011	Dep. Var.	Rain

Regression output
variables	coefficients	std. error	t (df=20)	p-value
Intercept	-51.6838	535.8344	-0.096	.9241
Year	0.0333	0.2692	0.124	.9029

Thus, by using the Excel, the test statistics and the P-value is as:

  $\begin{array}{l}P=0.9241\\ t=0.124\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

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

Thus, we conclude thatthese data do not suggest any change in annual rainfall as time passes. The regression equation from the above regression analysis is as:

  $\begin{array}{l}\widehat{y}=b_0+b_{1}x\\ $ \Rightarrow \widehat{R}\mathrm{ain}=-51.6838+0.0333\times \text{Year}$\end{array}$