Chapter 10.1, Problem 18E

(a)

To determine

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

Answer to Problem 18E

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 the proportion of mice in an area with additional acorns that are in breeding conditions larger than the proportion of mice in an area without additional acorns that are in breeding conditions or not.

So, the given claim that: the proportion is greater for the mice in the first area with the acorns.

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 mice in an area with additional acorns that are in breeding conditions.

  $p_2=$ the proportion of mice in an area without additional acorns that are in breeding conditions.

(b)

To determine

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

Answer to Problem 18E

All conditions are not met.

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=54\\ x_2=10\\ n_1=72\\ n_2=17\end{array}$

Conditions to be satisfied:

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

Random: It is not satisfied because the mice are already living in those forests before any treatments were applied and thus the mice were not randomly selected nor randomly assigned to a treatment.

Independent: It is satisfied because the $72$ mice in the first forest is less than $10\%$ of all mice in the first forest and the $17$ mice in the second forest is less than $10\%$ of all mice in the second forest.

Normal: It is not satisfied because there are $10$ successes and $17-10=7$ failures in the second forest, which 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 the hypotheses test for the population proportions for testing a claim.