Final answer:
The difference between the z-test for μ using rejection regions and the z-test using a P-value lies in the decision-making process. Rejection regions compare the z-score to a critical value, while the P-value approach calculates the actual probability of the observed test statistic, using the significance level to make the decision.
Step-by-step explanation:
The z-test for μ is a statistical test used to determine if there is a significant difference between a sample mean and a known population mean. The two approaches for making a decision in a z-test are using rejection regions (critical values) or using the P-value. In the rejection region approach, the z-score is compared to a critical value which is determined by the chosen significance level (α). If the z-score falls beyond this critical value in the rejection region, the null hypothesis (H₀) is rejected. On the other hand, when using the P-value approach, the actual level of significance of the z-score is calculated.
The P-value represents the probability of observing a test statistic as extreme as, or more extreme than the one calculated, assuming that the null hypothesis is true. If the P-value is less than the chosen significance level (α), then the null hypothesis is rejected. This approach provides the advantage of giving a direct measure of the evidence against the null hypothesis. In summary the main difference between using rejection regions and P-values is that the former involves a pre-determined critical value, while the latter provides a specific probability that measures the strength of the evidence against the null hypothesis.