Question

Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.6\) vs \(H_{a}: p>0.6\) Sample data: \(\hat{p}=52 / 80=0.65\) with \(n=80\)

Short answer

The null and alternative hypotheses are \(H_{0}: p=0.6\) and \(H_{a}: p>0.6\), respectively. The sample proportion is \(\hat{p}=52 / 80=0.65\) from \(n=80\) observations. A randomization distribution needs to be generated using software followed by the calculation of the p-value, which gives the chance of observing what we did if the null hypothesis was true.

Step-by-step solution

State the Hypotheses

The null hypothesis (\(H_{0}\)) is that the population proportion \(p\) equals 0.6, while the alternative hypothesis (\(H_{a}\)) is that the population proportion \(p\) is greater than 0.6. In other words, \(H_{0}: p=0.6\) and \(H_{a}: p>0.6\).

Analyze Sample Data

The sample data are given as a proportion: \(\hat{p} = 52/80 = 0.65\). This sample data will be used to conduct the hypothesis test.

Generate Randomization Distribution

Using StatKey or similar software, generate a randomization distribution. In StatKey, select 'Test for a Single Proportion', then 'Edit Data' to enter the sample information. Then, use the software to create the randomization distribution, which models what might happen if the null hypothesis is true.

Calculate p-value

Calculate the p-value, which is the probability under the null hypothesis of obtaining a result as extreme as, or more extreme than, the observed result. This p-value can be obtained by counting what proportion of the simulated randomization statistics are greater than or equal to the observed sample proportion of 0.65. The p-value is then this proportion.