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In any hypothesis test, the null hypothesis, denoted as \( H_0 \), represents the status quo or a default position that there is no difference or no effect. In the context of the presented exercise, the null hypothesis corresponds to the politician’s claim of receiving \(60\%\) of the vote, formally expressed as \( H_0 : p = 0.60 \). When performing the test, we initially assume the null hypothesis to be true and use statistical analysis to determine if there is sufficient evidence to reject it in favor of the alternative hypothesis.

It's crucial to frame the null hypothesis precisely since it defines the course of the testing procedure. If the evidence collected through a systematic method contradicts this hypothesis significantly, we consider it not likely to be true.

The alternative hypothesis, \( H_1 \), is what a researcher aims to prove. It suggests that there is a significant difference or effect. In our example, the alternative hypothesis is that the politician will not receive exactly \(60\%\) of the vote. This is written as \( H_1: p eq 0.60 \).

The alternative hypothesis is directly opposed to the null hypothesis, and it encompasses every other possible outcome that is not covered by the null hypothesis. The determination of whether the null hypothesis is rejected or not hinges on the gathered data and the statistical test applied to it. If the null hypothesis is rejected, we accept the alternative hypothesis, which indicates a significant difference detected by the test.

The p-value is the probability of observing data as extreme as, or more so, than what was actually observed, given that the null hypothesis is true. It is a measure used to decide whether to reject the null hypothesis.

To use the p-value approach, we compare the p-value to a pre-determined significance level, usually denoted as \(\alpha\). If the p-value is less than or equal to \(\alpha\), we have enough evidence to reject the null hypothesis. In our exercise, since the calculated p-value of \(0.0456\) is less than the level of significance \(0.05\), we reject the null hypothesis, indicating that the politician's claim is not supported by the data.

The test statistic is a numerical value calculated from sample data during a hypothesis test. It is a standardized value that is used for hypothesis testing. In the case of testing a proportion, the test statistic can be calculated using the formula: \[ Z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0}(1 - p_{0})}{n}}} \]

In this example, the test statistic is a Z-score, which measures the number of standard deviations a data point is from the population mean under the null hypothesis. With the given data (50 voters out of 100 in favor), the calculated Z-score is -2. This test statistic is later compared to values from the standard normal distribution to either support or reject the null hypothesis.

The level of significance, denoted as \(\alpha\), is the cutoff point that you choose to determine whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is true, but incorrectly rejected. A common choice for \(\alpha\) is \(0.05\) or \(5\%\).

The level of significance is set before examining the data, and it serves as a threshold for making our decision. For this exercise, the significance level of \(0.05\) means there is a \(5\%\) chance of concluding that the politician will not receive \(60\%\) of the votes when she actually will. Since the p-value in our test is below the defined significance level, we conclude that there is significant evidence to reject the null hypothesis at the \(5\%\) level of significance.