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The z-test is a statistical method used for hypothesis testing. It helps you decide if there is enough evidence to reject a null hypothesis. The test is especially useful when dealing with large sample sizes, typically greater than 30.

In a z-test, you compare the sample mean to a known value (like the population mean). You calculate the z-test statistic to see how the sample differs from what is expected if the null hypothesis were true. This test offers an effective way to judge the probability that an observed difference could have happened randomly.

The z-test formula is:

• \( z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \)

Here, \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, \( s \) is the sample standard deviation, and \( n \) is the sample size. Using these inputs, the formula provides a z-value, which helps you determine how far your sample mean is from the hypothesized population mean, expressed in standard deviation units.

The null hypothesis (denoted as \( H_0 \)) represents a default position that there is no effect or no difference. In the context of hypothesis testing, it proposes that any kind of difference or significance you see in a set of data is due to chance. It's the cornerstone of hypothesis testing since the main goal is to know whether to reject this baseline notion.

For example, in the exercise, the null hypothesis was \( H_0: \mu = 1.88 \). This hypothesis states that the mean turn-around time is exactly 1.88 days. When you conduct the test, you are trying to gather enough evidence to prove this hypothesis false or unable to be rejected.

In practice, if your test statistic indicates that the null hypothesis is likely not true, you may reject \( H_0 \) with some level of confidence. This allows for meaningful conclusions based on statistical evidence.

The alternative hypothesis (denoted as \( H_a \)) is your test or research hypothesis. It represents a statement that is believed to be true and one that you aim to support through evidence from the data. Unlike the null hypothesis, the alternative claims that there is an effect, a difference, or a change.

In the exercise, the alternative hypothesis was \( H_a: \mu > 1.88 \). This reflects that the mean turn-around time is greater than what was initially claimed.

When testing hypotheses, the goal is to see if there is enough statistical evidence to reject the null hypothesis in favor of the alternative hypothesis. The choice between the null and alternative depends on the test statistics and the evidence gathered during analysis. By establishing a clear alternative hypothesis, you focus your investigation on the possibility of an effect or difference being present.

The significance level, denoted by \( \alpha \), is a critical value that determines the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which occurs when you wrongly reject a true null hypothesis.

A common choice for \( \alpha \) is 0.05, but it can vary depending on the context of the study. In our exercise, the significance level was set at 10% (0.10). This means you are willing to accept a 10% risk of rejecting \( H_0 \) when it is actually true.

When you evaluate the test statistic, if it is beyond the critical value corresponding to your significance level, you reject the null hypothesis. Hence, the significance level is a measure of your confidence in the test results. Picking the right \( \alpha \) helps balance the risk of errors against the need for statistical certainty. In a right-tailed test, as in our case, the calculated critical z-value reflects this threshold needed to confirm statistically meaningful results.