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In hypothesis testing, the critical value is a threshold or boundary that divides the acceptance region from the rejection region for the null hypothesis. This value helps determine whether the test statistic falls within the rejection region, which would indicate that the null hypothesis should be rejected.

When performing a hypothesis test, you set a significance level, typically denoted as \( \alpha \). This determines how far from the mean of the distribution your test statistic must fall to reject the null hypothesis.

• For a one-tailed test, the critical value represents the point beyond which only \( \alpha \) percent of the data would lie.
• In this exercise, the critical value is determined by a \( z \)-distribution because we're comparing proportions.

For a one-tailed test with \( \alpha = 0.05 \), the critical \( z \)-value is found from standard normal distribution tables and is approximately \( 1.645 \). If your test statistic exceeds this value, you would reject the null hypothesis.

The test statistic is a standardized value that measures the degree of divergence between sample data and a null hypothesis. It's calculated using statistical formulas suited to the specific conditions of your hypothesis test. In the case of proportions, like this, it's determined using the formula for the difference between two sample proportions.

For the exercise provided, the formula used was:\[ z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]Where:

• \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions for women and men, respectively.
• \( \hat{p} \) is the pooled sample proportion calculated from both groups.

Once you perform the calculations, you get a test statistic that is then compared to the critical value. In this instance, the test statistic \( z \approx 1.06 \) did not exceed the critical value, suggesting the null hypothesis was not rejected.

Hypotheses are foundational to conducting hypothesis testing. They provide statements about population parameters that we aim to test statistically against our sample data.

• The **Null Hypothesis** \((H_0)\) represents a statement of no effect or no difference. In this problem, it states that the proportion of hypertensive women is less than or equal to that of men \( ( p_1 \leq p_2 ) \).
• The **Alternative Hypothesis** \((H_a)\) suggests otherwise; it is often what researchers hope to prove. Here, it claims that the proportion of women with hypertension is greater than that of men \( ( p_1 > p_2 ) \).

Identifying these hypotheses correctly is crucial as they guide the test's direction (one-tailed or two-tailed) and influence the determination of critical values and the interpretation of results.

The significance level, denoted as \( \alpha \), is the probability threshold used to determine when to reject the null hypothesis. It signifies the level of risk you're willing to take that you'll reject a true null hypothesis, often referred to as a Type I error.

• Commonly, a significance level of \(0.05\) is used, meaning there's a 5% chance of incorrectly rejecting the null hypothesis.
• In hypothesis testing, a smaller \( \alpha \) level reduces the risk of a Type I error but requires a stronger statistical signal to reject the null hypothesis.

In this test involving hypertension proportions, \( \alpha = 0.05 \) sets a conservative threshold, necessitating that results must be significant enough beyond this level to assert that women have a higher proportion with confidence. Here, the test did not exceed this threshold, causing retention of the null hypothesis and indicating insufficient evidence to support the claim.