Question

Null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1} \neq p_{2}\)

Short answer

The notation for a sample statistic in this case is \((\hat{p_1} - \hat{p_2})\). This represents the difference in sample proportions, which will be used to create the randomization distribution for the hypothesis test.

Step-by-step solution

Understand Hypothesis Testing

The first step is to understand that in a hypothesis test, we compare a null hypothesis (which is usually a claim of no effect, or no difference) against an alternative hypothesis (which usually claims an effect or difference). In this particular case, our null hypothesis is that both populations have the same proportion \(p_1 = p_2\), and our alternative hypothesis is that the proportions are not equal \(p_1 \neq p_2\).

Identify the Sample Statistic

We usually use sample statistics as an estimate of the corresponding population parameter. This lets us compare our observed data to what we would expect under the null hypothesis. In a test for proportions, this will usually be the sample proportion \(\hat{p}\). When comparing two proportions, we would usually use the difference in sample proportions \(\hat{p_1} - \hat{p_2}\).

Specify the Notation for the Sample Statistic

We now need to specify a notation for the sample statistic. In the case of comparing two proportions, we decided to use the difference in sample proportions, \(\hat{p_1} - \hat{p_2}\). So, the required notation for the sample statistic in this exercise is \((\hat{p_1} - \hat{p_2})\).