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When performing hypothesis testing for proportions, it's crucial to start by establishing a clear hypothesis. The null hypothesis (\(H_0\)) usually reflects the idea that there has been no change or that the parameter of interest remains at a specific value. In this case, our null hypothesis is \(H_0: p = 0.24\), meaning that 24% of workers do not need to contribute to their health care premiums.
This hypothesis is essential to test against the alternative hypothesis (\(H_a\)), which posits that the proportion has declined, stated as \(H_a: p < 0.24\). Creating these hypotheses is the foundation of hypothesis testing. It sets the stage for calculating various statistical measures and, ultimately, making a decision based on data. Always ensure the hypotheses reflect the question you are interested in testing, in this case, whether there has been a decline.

The point estimate is an essential tool that gives us a snapshot of the proportion we are testing. It's derived from the sample data and acts as an unbiased estimate of the population proportion. Here, the point estimate is calculated using the formula \( \hat{p} = \frac{x}{n} \), where \(x\) is the number of successes in our sample, and \(n\) is the total sample size.
In our exercise, 81 workers out of a sampled 400 do not contribute to the healthcare premium. Thus, the point estimate is \( \hat{p} = \frac{81}{400} = 0.2025 \).This means our best estimate, based on the sampled data, is that 20.25% of workers are not required to contribute. Understanding this estimate helps us gauge where the actual population proportion might lie.

The Z-statistic is a vital component when determining the statistical significance of an observed difference from the expected proportion. It measures how many standard deviations our point estimate is from the null hypothesis proportion. To calculate the Z-statistic, use the formula:\[ Z = \frac{\hat{p} - p_0}{SE} \]where \( \hat{p} \) is the point estimate, \(p_0\) is the proportion under the null hypothesis, and \(SE\) is the standard error of the sample proportion. The standard error is given by:\[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} \]In this exercise, our Z-statistic was calculated to be approximately -1.75.A Z-statistic helps in deciding how far and in what direction our sample proportion deviates from the null hypothesis proportion. It ultimately guides the decision to accept or reject the null hypothesis.

Statistical significance helps infer if an observed effect or difference in our study is genuine or due to random chance. It’s determined by comparing our calculated statistic to a critical value at a chosen significance level, \(\alpha\). In a hypothesis test, such as the one in this exercise, if the tested statistic (here, the Z-statistic) falls into the rejection region, it indicates statistical significance. For this test, we chose a significance level of \(\alpha = 0.05\), which implies a 5% risk of concluding that a difference exists when there is none. The critical Z-value for a left-tailed test at this level is approximately \(-1.645\). Since the calculated Z-statistic of \(-1.75\) is less than \(-1.645\), we determine there is a statistically significant decline in the proportion of workers not required to contribute. This means the evidence supports the conclusion that fewer workers aren't contributing compared to the past, beyond what could be expected by random variation alone.