We have given the value obtained for the test statistic, z,...

We have given the value obtained for the test statistic, $z,$ in a one-mean z-test. We have also specified whether the test is two tailed, left tailed, or right tailed. Determine the P-value in each case and decide whether, at the $5 \%$ significance level, the data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Two-tailed test: a. $z=-1.66$ b. $z=0.52$

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Key Concepts

One-Sample Z-Test

A one-sample z-test is used to determine whether a sample mean is significantly different from a known or hypothesized population mean. It assumes that the population standard deviation is known and employs the z-statistic to measure the distance between the sample mean and the null hypothesis mean in standard error units.

P-Value

The P-value represents the probability, under the null hypothesis, of observing a test statistic as extreme as or more extreme than the one computed from the sample data. It provides a measure of the evidence against the null hypothesis; a lower P-value indicates stronger evidence to reject the null hypothesis.

Two-Tailed Test

A two-tailed test considers deviations in both directions away from the hypothesized value. In such a test, the significance level is divided equally into the two tails of the distribution, and the P-value is computed as the combined probability of the observed test statistic falling in either extreme tail under the null hypothesis.

Significance Level

The significance level, often denoted by alpha (?), is the threshold for rejecting the null hypothesis. It is the maximum probability of committing a Type I error—rejecting a true null hypothesis—and is typically set at 5% in many hypothesis testing scenarios.

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