Chapter 20, Problem 28E

To determine

To comment and write up your own conclusions based on an appropriate confidence interval as well as a hypothesis test.

Answer to Problem 28E

We reject that the proportion of trees damaged by the acid rain in the Hopkins forest is $15\%$ and the confidence interval will be as: $(0.165,0.335)$ .

Explanation of Solution

It is given in the question that a study of the effects of acid rain on trees in the Hopkins Forest shows that $25\text{ of 100}$ trees sampled exhibited some sort of damage from acid rain. Thus, the null hypothesis states that the rate is equal to $15\%$ . And the alternative hypothesis states that the rate is more than $15\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.15\\ H_a:p>0.15\end{array}$

Thus, let us check the appropriate conditions and assumptions for inference to proceed further to test hypothesis as:

Random condition: It is satisfied as we assumed that the sample was randomly selected from the population.

Independent condition: It is satisfied as we can assume that the sample is independent.

  $10\%$ condition: It is satisfied as the sample is less than $10\%$ of all population.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=0.15(150)>10\\ n(1-p)=0.85(150)>10\end{array}$

Thus, all the conditions are satisfied. Thus, for this first we have to select $5:1-\text{PropZInt}$ from the STAT TESTS menu. Then, enter the observed count and sample size for the sample. Then we will indicate what kind of test we want: one-tail lower tail, two-tail, or one-tail upper tail. Then, we will specify whether to calculate the result or draw the result.

So, by putting the values above by using the calculator we will get the result as,

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=2.80\\ P-\text{value}=0.003\end{array}$

And as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

And now, to find the confidence interval we will use the calculator $\text{TI}-89$ . So, we have to first go to the $\text{Ints}$ menu and select $5:1-\text{PropZInt}$ . Then we have to enter the number of successes observed and the sample size. Then specify a confidence level. Then the interval will be calculated. Thus, the confidence interval will be as:

  $(0.165,0.335)$

Thus, we conclude that we reject that the proportion of trees damaged by the acid rain in the Hopkins forest is $15\%$ .