Chapter 10.1, Problem 17E

(a)

To determine

To state an appropriate hypotheses for performing a significance test and also define the parameters of interest.

Answer to Problem 17E

The hypotheses are as:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Explanation of Solution

It is given in the question that the researchers want to know that if there is a difference in the proportion of shrubs that received the treatment and resprouted and the shrubs that did not received the treatment and resprouted.

So, the given claim that: the proportion is greater the shrubs in the treatment group.

Now, we have to find out the appropriate hypotheses for performing a significance test.

Thus, the claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportions are equal. If the null hypothesis is the claim then the alternative hypothesis states the opposite of the null hypothesis.

Therefore, the appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Where we have,

  $p_1=$ the proportion of shrubs that received the treatment and resprouted.

  $p_2=$ the proportion of shrubs that did not received the treatment and resprouted.

(b)

To determine

To check if the conditions for performing the test are met or not.

Answer to Problem 17E

All conditions are not met.

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=12\\ x_2=8\\ n_1=12\\ n_2=12\end{array}$

Conditions to be satisfied:

There are three conditions to be satisfied for calculating a hypotheses test for the population proportions:

Random: It is satisfied because the treatments are independent random samples assigned to shrubs.

Independent: It is satisfied because assuming $12$ shrubs is less than $10\%$ of all shrubs.

Normal: It is not satisfied because there are $12$ successes and $12-12=0$ failures in the first sample while zero is not at least ten.

Thus, all the conditions are not satisfied since the large counts conditions is not satisfied and therefore it is not appropriate to use a hypothesis test for testing a claim.