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A proportion test is a statistical method used to analyze whether a sample proportion differs from a known or hypothesized population proportion. This kind of test is very helpful in understanding whether a characteristic or feature in a sample is characteristic of the entire population or if it is due to random chance. In this exercise, for example, the goal is to determine if the 62% proportion of cars passing emissions in a county is different from the known 70% statewide proportion.
The test involves comparing the sample proportion \(\hat{p}\) to the population proportion \(p_0\). If the difference is statistically significant, we can conclude that the sample provides evidence that the true population proportion differs from the hypothesized proportion. This test frequently uses a normal distribution to approximate the distribution of the sample proportions, especially when the sample size is large. This makes the proportion test a powerful tool in hypothesis testing, providing clear insights into whether observed differences are significant or not.

In hypothesis testing, establishing the null and alternative hypotheses is the first crucial step. The null hypothesis, denoted as \(H_0\), represents the status quo or a statement of no effect or no difference. It typically reflects the assumption that any kind of deviation from the expected norm is just due to random variation.
In the context of this exercise, the null hypothesis is that the proportion of cars passing emissions is the same as the previous statewide proportion, which is 70%. Thus, we state given \(H_0 : p = 0.70\).
The alternative hypothesis, denoted as \(H_a\), embodies the assertion that opposes the null hypothesis. It suggests that there is a significant difference that needs to be addressed. Here, \(H_a: p eq 0.70\) suggests that the proportion of cars passing emissions actually differs from the 70% proportion seen previously statewide. This two-tailed approach tests for deviations in both directions, indicating possible increases or decreases in the proportion.

The significance level, denoted by \(\alpha\), is a threshold set by the researcher to decide if the observed effect is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true - a type I error.
For this exercise, a significance level of \(\alpha = 0.05\) is used. This implies that there is a 5% risk of concluding there is a difference in proportions when actually, none exists. Choosing a significance level involves a trade-off between risk and confidence. A lower significance level means more confidence in results, but it might also increase the chance of missing a true effect (a type II error).
Commonly used \(\alpha\) values include 0.01, 0.05, and 0.10. The critical values derived from these \(\alpha\) levels help determine the rejection region for the null hypothesis. In a two-tailed test with \(\alpha = 0.05\), the critical z-values from the standard normal distribution are approximately -1.96 and 1.96. Observing a test statistic beyond these critical values leads to the rejection of the null hypothesis.

Calculating the test statistic involves applying a specific formula to determine how far the sample proportion is from the hypothesized population proportion in terms of standard deviations. For a proportion test, the formula for the test statistic \(z\) is given by:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \]where

• \(\hat{p}\) is the sample proportion
• \(p_0\) is the hypothesized population proportion
• \(n\) is the sample size

In this exercise:

• The sample proportion \(\hat{p}\) is 0.62
• The hypothesized proportion \(p_0\) is 0.70
• The sample size \(n\) is 200

By substituting these values into the formula, the test statistic is calculated as approximately \(-2.83\). This value indicates how many standard deviations the observed proportion is from the hypothesized proportion. Since this test statistic falls into the rejection region (beyond the critical values of -1.96 and 1.96), we conclude that the sample provides sufficient evidence against the null hypothesis, suggesting a real difference in proportions.