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Proportion hypothesis tests are an essential part of statistics when dealing with questions about population proportions. You use these tests to make an informed guess about the population proportion based on sample data. It starts with defining two hypotheses:

• The null hypothesis (H_0): This is a statement of no effect or no difference. In our exercise, this hypothesis is that the college students with hypertension proportion is equal to a specified value, 9.2%.
• The alternative hypothesis (H_1): This statement counters the null hypothesis. For the exercise, we ask if the proportion is greater than 9.2%, indicating a right-tailed test.

After stating these hypotheses, the test aims to determine how likely it is that the sample data happened by random chance if the null hypothesis is true. This involves calculating a test statistic, which in our case refers to how much the sample proportion differs from the hypothesized population proportion. Then, by comparing this value to a known threshold or critical value, you decide whether to reject the null hypothesis or not.

The Z-test for proportions is a statistical test used to determine whether there is a significant difference between an observed sample proportion and a known population proportion. Here, this test involves the calculation of a "z-statistic."To calculate this z-statistic, we need certain elements: the sample proportion (\hat{p}), the hypothesized population proportion (p_0), and the sample size (n). The formula is straightforward:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

• The sample proportion \hat{p} is calculated by dividing the number of successful instances by the total sample size; in this case, 29 students out of 196 had hypertension.
• The target proportion, p_0, represents the null hypothesis: 9.2% or 0.092.

After plugging these values into the equation, the z-statistic calculated gives insight into how far and in which direction our sample proportion is from the null hypothesis, measured in terms of standard deviation units. This value is crucial for comparing against the critical value to draw a conclusion.

The significance level, often denoted as \( \alpha \), represents the probability of rejecting the null hypothesis when it is actually true. It helps define the threshold for making this critical decision. In hypothesis testing, a common level of significance is 5% (or \( \alpha = 0.05 \)).When you perform a hypothesis test like our example, you compare the calculated test statistic with a critical value corresponding to the chosen significance level. The critical value depends on the type of test (right-tailed, left-tailed, or two-tailed).

• In a right-tailed test as in the exercise, the setup implies interest in whether the sample statistic exceeds the hypothesized population value.
• For \( \alpha = 0.05 \), the critical value from a standard normal distribution (z-distribution) table is approximately 1.645.

The decision rule is straightforward: if the calculated z-statistic is greater than the critical value, you reject the null hypothesis. Our example showed a z-statistic of 2.447, indicating that at a 5% significance level, there was enough evidence to reject the null hypothesis and conclude that students' hypertension rates during exam week were indeed higher."