91Ó°ÊÓ

A proportion test is a statistical method used to compare a sample proportion to a known population proportion. It helps determine whether there is a significant difference between the two. In the context of the exercise, we want to find out if the current proportion of workers who do not have to contribute to their health care plan has decreased from the previous 24%. This is done using a one-sample Z-test for proportions.

The Z-test calculates how many standard deviations our sample proportion is from the population proportion. If this difference is too large, it implies that the current proportion is likely different from the historical one.

To conduct this test, you'll need:

• The sample proportion, which is the ratio of workers in the study not required to contribute to their health care plan.
• The known population proportion, which in this example is 24% or 0.24.
• The sample size, which is the total number of workers in the study.

Performing this test provides a statistical foundation to determine if any observed changes are genuine or just due to random sampling variation.

In hypothesis testing, formulating null and alternative hypotheses is the initial step. The null hypothesis ( H_0) is a statement of no effect or no difference. It's what you assume to be true before analyzing the data. For this problem, the null hypothesis is that the proportion of workers not required to contribute remains at 24%.

Conversely, the alternative hypothesis ( H_1) represents what you seek to prove. It's a claim about the population parameter that suggests a change or effect. In this scenario, the alternative hypothesis claims that the proportion has decreased from 24%, stated as H_1: p < 0.24.

Understanding these hypotheses is essential as the entire procedure of testing and conclusions revolves around them.

• H_0: p = 0.24 (No change in proportion).
• H_1: p < 0.24 (Proportion has decreased).

Testing these hypotheses through a suitable statistical test helps in making data-driven decisions.

The point estimate is a single value that provides the best estimate of the population parameter based on the sample data. In this exercise, it estimates the proportion of workers who did not pay for their health care plan. The point estimate is calculated through the formula: \(\hat{p} = \frac{x}{n}\)

Here, \(x\) is the number of workers who did not have to contribute, in this case, 81, and \(n\) is the total number of workers surveyed, which is 400. Plugging these numbers into the formula gives the point estimate:\[\hat{p} = \frac{81}{400} = 0.2025\]

This means that, according to the sample data, about 20.25% of workers were not required to contribute to their healthcare plan. This point estimate is used for further testing to see if there is statistical evidence suggesting a change from the original 24% figure.

The significance level, denoted by \(\alpha\), is a crucial part of hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, commonly known as the Type I error. In other words, it's the risk you're willing to take when making a claim about the population based on your sample data.

In this exercise, the significance level is set at \(\alpha = 0.05\). This implies that you are accepting a 5% chance of concluding that there is a decline in the proportion when there isn't one.

Choosing the right significance level is important. A higher \(\alpha\) increases the probability of a Type I error, while a lower \(\alpha\) might make the test too conservative. Always consider the context when setting your significance level. For example, more critical decisions may require a lower \(\alpha\), while others may tolerate higher ones.