Chapter 10.1, Problem 16E

(a)

To determine

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

Answer to Problem 16E

The hypotheses are as:

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

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 all Illinois high school freshmen and seniors who have used anabolic steroids or not.

So, the given claim that: difference between the proportions.

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:

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

Where we have,

  $p_1=$ the proportion of high school freshmen in Illinois that used anabolic steroids.

  $p_2=$ the proportion of high school seniors in Illinois that used anabolic steroids.

(b)

To determine

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

Answer to Problem 16E

All conditions are met.

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=34\\ x_2=24\\ n_1=1679\\ n_2=1366\end{array}$

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 samples are independent random samples from separate populations.

Independent: It is satisfied because the $1679$ high school freshmen is less than $10\%$ of all high school freshmen and the $1366$ high school seniors is less than $10\%$ of all high school seniors.

Normal: It is satisfied because there are $34$ successes and $1679-34=1645$ failures in the first sample and there are $24$ successes and $1366-24=1342$ failures in the second sample, which are all at least ten.

Thus, all the conditions are satisfied and it is appropriate to use the hypotheses test for the population proportions for testing a claim.