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The **z-test for proportions** is a statistical tool used to compare observed sample proportions to a known population proportion. In other words, it's a way to determine if there is a significant difference between a sample proportion and a specific known value. This test is applicable when you have a binary outcome, such as yes or no, vote or not vote.

In the exercise, a z-test for proportions is used to check if the proportion of adult Americans who always vote, as claimed by Time magazine (35%), matches with our sample data. To do this:

• First, calculate the sample proportion (\(\hat{p}\)), which is the number of successes (in our exercise, the number of people who always vote) divided by the total sample size.
• The z-score then measures how many standard deviations the observed sample proportion is away from the hypothesized population proportion, using:\[z = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]

A significant z-score indicates a difference between the sample proportion and the claimed population proportion.

When conducting a hypothesis test, you start by stating the **null and alternative hypotheses**. These hypotheses are statements about a population parameter (typically a proportion or a mean) that you want to test. The null hypothesis (\(H_0\)) is a statement that there is no effect or no difference, while the alternative hypothesis (\(H_a\)) represents what you want to prove or the presence of an effect.

For the problem at hand, the null hypothesis is that the proportion of adult Americans who always vote is equal to 35%, as reported by Time. In symbol form, it's:\[H_0: p = 0.35\]The alternative hypothesis is that the actual proportion is different from this reported value. This two-tailed hypothesis is expressed as:\[H_a: p eq 0.35\]Choosing between these hypotheses involves statistical testing to see if there is enough evidence to support the alternative over the null. In hypothesis testing, failing to reject the null does not confirm it, but rather indicates insufficient evidence to support the alternative.

The **significance level** in hypothesis testing, denoted by \(\alpha\), is the threshold used to determine whether the null hypothesis should be rejected. It is a pre-determined probability of making a Type I error - rejecting a true null hypothesis.

In practice, it indicates how willing we are to risk being wrong if we claim there is an effect or a difference when there isn't one. Common significance levels are 0.05, 0.01, and 0.10. In this exercise, the significance level is set at 0.01, meaning there's a 1% risk we wrongly reject the null hypothesis.

When using this significance level in a two-tailed z-test, as in the exercise, you identify the critical z-values that frame the rejection region. For \(\alpha = 0.01\), these critical values in a standard normal distribution would be \(Z_{\alpha/2} = \pm 2.576\). These cutoffs indicate the points beyond which the null hypothesis is rejected if the test statistic falls outside this range.

Setting up a proper significance level helps make informed decisions based on your hypothesis test outcomes.