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The z-test is a statistical method used to determine if there is a significant difference between the sample mean and a known population mean. It's particularly useful when the sample size is large (usually above 30), as in the exercise where 500 samples were observed. The z-test employs the standard normal distribution to evaluate the test statistic against critical values.
The formula for the z-test is given by:\[ z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \]Where:

• \( \bar{x} \) is the sample mean
• \( \mu_0 \) is the population mean according to the null hypothesis
• \( s \) is the sample standard deviation
• \( n \) is the sample size

In our exercise, the z-test helps to determine whether the observed average length of hospital stays (5.4 days) significantly exceeds the hypothesized average of 5 days.

The null hypothesis is a crucial component in hypothesis testing. It proposes that there is no effect or no difference, and it sets up a baseline that we test against. Essentially, it's a statement of no change or no effect. In our exercise, the null hypothesis \( H_0: \mu = 5 \) asserts that the average hospital stay is exactly 5 days.Accepting the null hypothesis implies there's not enough evidence to claim a difference from the stated mean. It acts as a default assumption unless the evidence strongly suggests otherwise. Null hypotheses are always framed to allow for their refutation, which is the aim of most hypothesis tests. In statistical testing, failing to reject the null does not prove it true, just as rejecting it does not prove the alternative hypothesis true.

The alternative hypothesis is what you suspect might be true or aim to support with evidence. It stands in contrast to the null hypothesis. For the exercise, the alternative hypothesis \( H_a: \mu > 5 \) suggests that the average stay in hospitals is more than 5 days, in line with the federal agency's hypothesis.The alternative hypothesis is crucial because it identifies what is being tested and what the research aims to demonstrate. If the data provides sufficient evidence against the null hypothesis, we accept the alternative hypothesis. In practice, the alternative hypothesis is what you're trying to prove. It directs the nature of the test being performed—in this case, a right-tailed test, which checks for evidence that the sample mean is greater than the hypothesized mean.

The significance level, denoted by \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true. Essentially, it's the threshold for determining whether the observed test statistic is extreme enough to warrant rejecting the null hypothesis.
In most tests, a significance level of 0.05 is commonly used, as seen in our exercise. This means there's a 5% risk of mistakenly claiming that the sample provides enough evidence to reject the null hypothesis when it's actually correct.
A lower significance level means a stricter criterion for rejecting the null hypothesis, minimizing the risk of a Type I error (false positive). In statistical tests, setting the significance level is an important decision that reflects the acceptable risk of making such an error. For our exercise, since the calculated z-value exceeded the critical value at \( \alpha = 0.05 \), the null hypothesis was rejected.