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When conducting a hypothesis test, one of the first steps is determining whether to use a one-tailed or two-tailed test. In this scenario, the alternative hypothesis is that the population mean is greater than 20, represented as \(H_{1}: \mu > 20\). This indicates we are only interested in deviations in one direction (greater than 20). Thus, this is known as a one-tailed test.

A one-tailed test checks if the parameter falls only in one direction from the hypothesized value. It is more powerful for detecting an effect in that specific direction compared to a two-tailed test, which considers deviations on both sides.

If the direction was not specified, or we were interested in deviations on both sides, we would use a two-tailed test. However, here, we have a clear direction defined in the alternative hypothesis, confirming it's a one-tailed test.

The test statistic is a crucial part of hypothesis testing. It's a standardized value that helps compare your sample results against the null hypothesis. For a population mean, when the population standard deviation is known, we use the formula for the Z-statistic:

\[ Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \]

Here, the sample mean is \(\bar{x} = 21\), the hypothesized mean \(\mu_0 = 20\), the population standard deviation \(\sigma = 5\), and the sample size \(n = 36\).

Plugging these values into the formula gives us a test statistic \(Z \approx 1.20\). This statistic tells us how many standard deviations the sample mean is above the hypothesized mean. A larger Z would suggest more evidence against the null hypothesis, but here, the calculated Z of 1.20 is not enough on its own to reject \(H_0\).

The decision rule in hypothesis testing is essential for determining whether to accept or reject the null hypothesis. It uses the test statistic and the significance level (\(\alpha\)) to make this decision.

For a one-tailed test at \(\alpha = 0.05\), the critical Z-value is 1.645. If the calculated test statistic is greater than this critical value, we reject the null hypothesis. In our example, the calculated Z is 1.20, which is less than 1.645.

This decision rule helps maintain the integrity of the test by setting a threshold for error. It minimizes the chances of making a Type I error (incorrectly rejecting a true null hypothesis).

Since 1.20 is not greater than 1.645, we adhere to the decision rule and do not reject the null hypothesis.

The p-value is a probability that provides an alternative way to make decisions in hypothesis testing. It measures how extreme the test statistic is, under the assumption that the null hypothesis is true.

For the calculated Z of 1.20, the p-value is approximately 0.1151. This figure represents the probability of observing a sample mean as extreme as 21 or more, assuming the null hypothesis \(H_0\) is true.

A p-value is compared to the significance level \(\alpha\) to decide on rejecting \(H_0\). If the p-value is less than or equal to \(\alpha\), we reject \(H_0\); otherwise, we do not reject it.

In this case, the p-value (0.1151) is greater than \(\alpha = 0.05\). This means we do not have enough evidence to reject \(H_0\). Simply put, there's not enough statistical proof to say that the population mean is greater than 20.