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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 the population mean you are testing against. It's particularly useful when you know the population standard deviation and have a sufficiently large sample size (usually over 30). In this scenario, we're testing whether the sample mean of 19.4 indicates that the population mean is less than 20. Here, you compute a test statistic using the formula:

• \( Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \)

In the formula:

• \(\bar{x}\) is the sample mean,
• \(\mu_0\) is the hypothesized population mean,
• \(\sigma\) is the population standard deviation, and
• \(n\) is the sample size.

For our exercise, using the given values, the Z-test statistic calculated was \( Z = -2.12 \).

The p-value is a critical component in hypothesis testing, providing the probability of observing a test statistic as extreme as the one obtained, under the assumption that the null hypothesis is true. In simpler terms, it helps us understand if our observed data can occur by random chance.
For the given Z-test statistic of -2.12 and considering it is a one-tailed test (since the alternative hypothesis is \( \mu < 20 \)), the p-value is the probability that the Z-score is less than -2.12.
Using Z-tables or statistical software, you can find that the p-value is approximately 0.0170. This small value indicates that such a result is unlikely to occur if the null hypothesis were true, thus suggesting significant evidence against it.

When discussing hypothesis testing, the critical value is a threshold that determines whether the null hypothesis should be rejected. It corresponds to the significance level \( \alpha \), which represents the probability of rejecting the null hypothesis when it is true. In our one-tailed test, the critical value is found on a Z-table for \( \alpha = 0.05 \).
For this test, the critical value is -1.645. This means that any Z-test statistic less than -1.645 will fall into the rejection region. With our calculated \( Z = -2.12 \), we see that it is indeed less than -1.645, indicating that our sample provides sufficient evidence to reject the null hypothesis.

The rejection rule provides the guideline for deciding whether to reject the null hypothesis in hypothesis testing. For this example, there are two methods: using the p-value and using the critical value.
The p-value approach involves comparing the p-value to the significance level \( \alpha \). If the p-value is less than \( \alpha \), we reject the null hypothesis. In our case, since 0.0170 is less than 0.05, we reject \( H_0 \).
The critical value approach uses the critical value to determine the rejection region. If the test statistic falls within this region, we reject \( H_0 \). With a critical value of -1.645 and a test statistic of -2.12, we again find ourselves in the rejection region.
Both methods lead to the same conclusion: there is enough evidence to reject the null hypothesis, supporting the claim that the population mean \( \mu \) is indeed less than 20.