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: \(H_{0}: p=0.7\) vs \(H_{a}: p<0.7\) Sample data: \(\hat{p}=125 / 200=0.625\) with \(n=200\)

Short answer

Using the given null and alternative hypotheses and the sample data, a randomization distribution was generated via StatKey. The obtained p-value would then be used to either reject or fail to reject the null hypothesis. For instance, a p-value less than 0.05 would lead to rejecting \(H_{0}\), providing strong evidence for the alternative hypothesis that the population proportion is less than 0.7.

Step-by-step solution

Calculate Sample Proportion

Sample proportion (\(\hat{p}\)) is calculated by dividing the number of successful outcomes by the total number of trials. In this case, \(\hat{p} = 125 / 200 = 0.625\).

Using StatKey

To use StatKey, firstly select 'Proportion' under 'Randomization Test' since the given data deals with proportions. Then Edit Data and enter the hypothesized population proportion from \(H_{0} (0.7)\) and the results obtained from the sample data, in this case, successes = 125 , sample size = 200. Then click 'Calculate'.

Generate Randomization Distribution and Calculate p-value

After results are entered, StatKey will generate a simulation to create a randomization distribution. The p-value is calculated as the proportion of simulated results that are greater than or equal to the observed sample proportion.

Interpret the p-value

The p-value represents the chance of getting the sample data if the null hypothesis is true. If the p-value is low (say less than 0.05), the null hypothesis will be rejected, indicating that the sample data provides strong evidence that the actual population proportion is less than 0.7. If the p-value is high, the null hypothesis will not be rejected, meaning there is not enough evidence to suggest the actual population proportion is less than 0.7.

Key concepts

Population Proportion

The concept of 'population proportion' is critical in statistics as it represents the fraction of individuals in a population that have a particular attribute or characteristic. For instance, it could be the proportion of voters in a country who support a particular policy, or the proportion of consumers who prefer a specific brand of coffee.

To put it into context, if you have a town where 70 out of 100 households have solar panels, the population proportion of households with solar panels is 0.7. In statistical hypothesis testing, we often use population proportion when we're trying to draw conclusions about a population based on a sample from that population.

P-value

Understanding the 'p-value' can be a bit tricky, but it's essentially a measure of the strength of the evidence against the null hypothesis provided by the data. It tells us how likely it is to observe a sample statistic as extreme as the test statistic, under the assumption that the null hypothesis is true.

If the p-value is low, it suggests that such an extreme observed outcome is unlikely under the null hypothesis, pointing towards the alternative hypothesis. However, a high p-value indicates that the observed data is relatively normal under the null hypothesis, and thus, does not provide strong evidence against it. It's common to use a threshold (like 0.05) to decide if a p-value is 'low' or 'high'.

Randomization Distribution

The 'randomization distribution' plays a pivotal role in understanding how unusual our sample results are, within the context of the null hypothesis. It's created by simulating many samples, assuming the null hypothesis is true, and then seeing how the sample statistics (like sample proportions) are spread out.

This distribution helps us to place our actual observed sample statistic. If our observed statistic falls far into the tail of this distribution, it's a signal that our result is unusual — and perhaps the null hypothesis isn't the best explanation.

StatKey

When it comes to making statistical concepts practical, 'StatKey' is a fantastic tool. This is software designed specifically for teaching and learning statistics. It simulates the randomization test for a single proportion, among other tests. Students can enter their sample data, like the number of successes and the total number of trials, to see visual simulations of distributions and to calculate relevant statistics, such as the p-value, for making informed decisions on hypotheses.

Sample Proportion

Finally, let's talk about 'sample proportion'. This is similar to population proportion, but it deals with the sample drawn from the population rather than the entire population. It's the fraction of the sample that exhibits the characteristic we're interested in.

In our example, where 125 out of 200 people prefer a certain beverage, the sample proportion is 0.625. This is a crucial piece of the puzzle because it's the actual measurement we compare against what we expect from the population, to draw conclusions and test hypotheses.