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In hypothesis testing, a one-tailed test is used when we are interested in determining if a parameter is significantly greater than or less than a certain value. In the given exercise, the alternative hypothesis tests if the mean is less than 220, which makes this a one-tailed test. Specifically, it is a left-tailed test because the direction is toward the lower tail of the distribution. This approach allows us to focus on one end of the distribution for more precise testing. A practical outcome of a one-tailed test is that it offers more power to detect an effect in one direction, as opposed to a two-tailed test, which considers deviations in both directions.

One-tailed tests are useful in instances where the outcome can only logically occur in one direction, or when such an outcome holds particular interest or significance to the researcher. By focusing on one direction, one-tailed tests enable researchers to make stronger inferences about the presence of an effect.

The critical value in hypothesis testing is the threshold against which the test statistic is compared to decide whether to reject the null hypothesis. For a one-tailed test at a 0.025 significance level, you find this value on a standard normal distribution table. In this case, the critical value for a left-tailed test is approximately -1.96.

The critical value is pivotal because it marks the point beyond which we consider the observed test statistic as unlikely under the null hypothesis. When the calculated test statistic falls beyond the critical value, we have sufficient evidence to reject the null hypothesis in favor of the alternative.

Understanding how to find and use the critical value is essential as it directly influences decision-making in statistical analysis by establishing a clear boundary for interpreting results.

The test statistic is a standardized value used to compare against the critical value to determine the strength of the evidence against the null hypothesis. For means with a known population standard deviation, the test statistic is calculated using:

\[z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\]

• \(\bar{x}\) is the sample mean.
• \(\mu\) represents the hypothesized population mean under \(H_0\).
• \(\sigma\) is the population standard deviation.
• \(n\) is the sample size.

In the exercise, substituting in the values gives \(z = -2.67\). This test statistic tells us how many standard deviations the sample mean is from the hypothesized population mean.

The sign and magnitude of the test statistic are vital: the sign indicates the direction of the sample mean relative to the hypothesized mean, while the magnitude suggests how typical or atypical the sample is under the null hypothesis.

The p-value is a probability that measures the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true.

In this context, for a \(z\) test statistic of \(-2.67\), the p-value is approximately 0.0038. This is much smaller than the 0.025 significance level used in the test, indicating strong evidence against the null hypothesis.

Interpreting a p-value is straightforward:

• A small p-value (typically \(<\) significance level) suggests rejecting \(H_0\).
• A large p-value indicates insufficient evidence to reject \(H_0\).

Thus, a p-value provides a quantitative measure for deciding whether to reject the null hypothesis. The smaller the p-value, the stronger the evidence against \(H_0\).

In hypothesis testing, the null hypothesis \((H_0)\) is a statement of no effect or no difference that serves as the starting point for statistical testing. It is the hypothesis that researchers typically aim to test against. In the exercise, \(H_0: \mu \geq 220\) suggests that the population mean is greater than or equal to 220.

The test aims to determine if there is enough statistical evidence to reject \(H_0\) in favor of the alternative hypothesis \((H_1)\), which in this case suggests \(\mu < 220\).

Null hypotheses are formulated to be straightforward and falsifiable. They provide a specific claim that can be tested with observed data, allowing researchers to use statistical methods to deduce if deviations from the claim are significant. It's important to note that rejecting \(H_0\) does not prove \(H_1\); it merely indicates that the data provide strong enough evidence against \(H_0\).