91Ó°ÊÓ

In hypothesis testing, the null hypothesis is a fundamental concept. It is a statement made in order to test if there is enough evidence to prove a certain claim wrong. For your exercise, the null hypothesis suggests that 50% of voters in Pennsylvania have voted for an independent candidate in past elections.
Simply put, the null hypothesis is a default position that there is no effect or no difference. It's what you assume to be true until the data tells you otherwise. If data shows enough evidence, the null hypothesis can be rejected. But if not, it remains in position, indicating no change from the expected distribution.
When a claim like the one in the exercise is made, statistical tests aim to either refute or fail to refute this claim. It's crucial to begin with a clear null hypothesis as it forms the basis for any hypothesis testing process.

Population proportion is a statistical measure. It represents the fraction of the total population that has a certain characteristic. In your exercise scenario, the population proportion refers to the percentage of all voters in Pennsylvania who voted for an independent candidate.
Represented by the symbol \(p\), determining this proportion isn't always straightforward. We often rely on sampled data to make inferences about the broader population. Here, the politician believes that \(p\), the hypothesized population proportion, is 0.5 or 50%.
Understanding the population proportion is key because it forms the foundation of the hypothesis. In this context, 50% is what the test will be comparing the sample results against to see if there's a significant difference.

Sample proportion is another pivotal concept in the context of hypothesis testing. It is determined from the sample data collected. The sample proportion, denoted as \(\hat{p}\), gives an estimate of the actual population proportion based on your sample.
In the exercise, out of 20 surveyed people, 12 reported voting for an independent candidate. Therefore, \(\hat{p} = \frac{12}{20} = 0.6\). This means 60% of the sample population voted for an independent candidate.
The sample proportion provides firsthand data and acts as a comparison point against the null hypothesis. It's the observed outcome in your surveyed group, which will be scrutinized in the hypothesis test.

The test statistic is a calculated value from your sample data used in hypothesis testing. It essentially measures how far the sample proportion is from the hypothesized population proportion. The formula given in your exercise is \( z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \).
Here, \( \hat{p} \) is the sample proportion, \( p \) is the hypothesized population proportion, and \( n \) is the sample size. By calculating this, we measure the variability of observations and compare it against what's expected if the null hypothesis holds true.
In the given solution, when you substitute your values into this equation, you end up with a test statistic of approximately 1.41. This value indicates how much the sample proportion deviates from the population proportion, given the size of your sample. The test statistic is critical for determining if the deviation observed is due to random chance or suggests a different conclusion.