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A z-test is a statistical method used when you want to test hypotheses about a population mean when the sample size is large and the population standard deviation is known. In the context of hypothesis testing, the z-test compares the sample mean to the population mean and checks how many standard deviations away the sample mean is from the hypothesized population mean.
The formula for a z-test is:

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

Where:

• \(\bar{x}\) is the sample mean
• \(\mu\) is the population mean
• \(s\) is the sample standard deviation
• \(n\) is the sample size

It is essential to use a z-test when a sample size is large, like in our exercise with a sample of 115 males, as this helps achieve more reliable results.

In hypothesis testing, the null hypothesis, usually denoted as \(H_0\), is a statement that there is no effect or no difference, and it is the hypothesis that researchers typically try to disprove. In our zinc intake example, the null hypothesis is that the average zinc intake is equal to the recommended 15 mg/day. Symbolically, it is expressed as:

• \(H_0: \mu = 15 \)

The objective is often to use data evidence to statistically reject \(H_0\), thereby allowing for the possibility that an alternative hypothesis (where a difference or effect exists) may be true. By rejecting the null hypothesis, conclusions can be drawn with a higher level of confidence.

The significance level, denoted as \(\alpha\), determines the probability threshold at which you reject the null hypothesis. Common choices for \(\alpha\) include 0.05, 0.01, or 0.10, with 0.05 being most common. This means you are willing to accept a 5% chance of rejecting the null hypothesis when it is actually true.
The smaller the significance level, the stricter the test is for deciding whether to reject the null hypothesis. In our example, with \(\alpha = 0.05\), we are using a standard threshold, which balances between obtaining evidence against \(H_0\) and not rejecting it too easily. The choice of \(\alpha\) can affect the likelihood of Type I errors (falsely rejecting a true \(H_0\)).

A critical value is the threshold that the test statistic must exceed in order to reject the null hypothesis. In a z-test, this value comes from a standard normal distribution table, determined by the chosen significance level.
For a one-tailed test at \(\alpha = 0.05\), a critical value is found at approximately -1.645 if you are testing for a value less than the hypothesized mean. This means if our calculated z-test statistic is less than -1.645, we reject \(H_0\).
In our zinc intake example, the test statistic calculated was -6.12, which is less than -1.645; therefore, it falls into the rejection region, which leads to rejecting the null hypothesis. Understanding such critical values helps us make informed decisions regarding our hypotheses.