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Hypothesis testing begins with establishing two conflicting statements: the null hypothesis and the alternative hypothesis. These hypotheses form the basis for your test. The null hypothesis, denoted as \( H_0 \), generally represents a statement of no effect or no difference. In this exercise, \( H_0 : p = 0.70 \) asserts that the proportion of adults not exercising regularly matches the reported 70%.

On the other hand, the alternative hypothesis, denoted by \( H_a \), stands for the statement we aim to provide evidence for. Here, \( H_a : p eq 0.70 \) suggests that the real proportion of non-exercising adults is different from 70%. By conducting the hypothesis test, the researcher aims to see if there's substantial evidence that the actual state-specific proportions differ from the national estimate.

A proportion test is used to compare an observed proportion from sample data to a claimed population proportion. It's particularly useful in studies like the one described here, where you need to evaluate whether a sample proportion significantly deviates from a known value.

The procedure involves calculating the sample proportion—the ratio of a particular category to the total number. This step was crucial for both Wisconsin and California. For example, Wisconsin had a sample proportion \( \hat{p}_{\text{WI}} = 0.72 \) as 252 out of 350 adults did not exercise. This proportion gives an experimental perspective on how real-world data may align or vary from theoretical claims. Through a proportion test, these sample figures are statistically compared to the benchmark of 70% to see if observed differences arise from chance or reflect true deviations.

The significance level, denoted \( \alpha \), represents the probability of rejecting the null hypothesis when it is actually true, commonly set at 0.05 for a 5% risk level. This threshold dictates the strictness of the test criteria. A 5% level means that if we did this test 100 times, we expect to mistakenly reject the true null hypothesis 5 times.

In hypothesis testing, such as in our state exercise comparison, the significance level determines the critical values of the test statistic. Specifically, it helps to draw boundary lines, often represented as z-values in a standard normal distribution. These critical values, like \( \pm 1.96 \) for a 95% confidence interval, determine the acceptance or rejection of \( H_0 \). If the test statistic falls beyond these key limits, the results are statistically significant, prompting a reconsideration of the null hypothesis.

A test statistic quantifies the extent to which your sample data departs from the expected under the null hypothesis. For the proportion test, the test statistic is computed using:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\]Here, \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized population proportion, and \( n \) is the sample size. This formula generates a z-score, representing how many standard deviations your sample outcome is from the referenced population proportion.

In our exercise, Wisconsin's calculated z-value was approximately 0.816, showing a slight deviation, not enough to shift beyond the critical z-limits. Contrarily, California's z-value was -2.592, indicating a pronounced deviation from 70%, prompting the rejection of \( H_0 \) there. Test statistics serve as fundamental tools to gauge whether sample observations are consistent with the null hypothesis or hint at more profound population dynamics.