Chapter 10.1, Problem 15E

(a)

To determine

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

Answer to Problem 15E

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 a greater proportion of $6\text{ to }7$ year olds can sort correctly than $4\text{ to }5$ year olds or not.

So, the given claim that: proportion is greater for the $6\text{ to }7$ year olds.

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 $4\text{ to }5$ year olds who can sort correctly.

  $p_2=$ the proportion of $6\text{ to }7$ year olds who can sort correctly.

(b)

To determine

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

Answer to Problem 15E

All conditions are met.

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=10\\ x_2=28\\ n_1=50\\ n_2=53\end{array}$

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 children are independent random samples.

Independent: It is satisfied because the $50$

  $4\text{ to }5$ year-olds is less than $10\%$ of all $4\text{ to }5$ year-olds and the $53$

  $6\text{ to }7$ year-olds is less than $10\%$ of all $6\text{ to }7$ year-olds.

Normal: It is satisfied because there are $10$ successes and $50-10=40$ failures in the first sample and there are $28$ successes and $53-28=25$ 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.