Explain the difference between the $z$-test for $\mu$ using rejection region(s) and the $z$-test for $\mu$ using a P-value

Choose the correct answer below
A. The $z$-test using rejection region(s) is used when the population is nomal. The $z$-test using a P-value is used when the population is not normal.
B. In the $z$-test using rejection region(s), the test statistic is compared with critical values. The $z$-test using a P-value compares the $P$-value with the level of significance $\alpha$
C. The z-test using rejection region(s) is used when the population is not normal. The $z$-test using a P-value is used when the population is normal
D. In the $z$-test using rejection region(s), the test statistic is compared with the level of significance $\alpha$. The $z$-test using a P-value compares the $P$-value with the critical values.
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Explain the difference between the $z$-test for $\mu$ using rejection region(s) and
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Step 1 ：The question is asking about the difference between two types of z-tests: one that uses rejection regions and one that uses a P-value. The z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. The test statistic is a z-score (z). The rejection region(s) approach involves comparing this test statistic with critical values, while the P-value approach involves comparing the P-value with the level of significance (α).

Step 2 ：The options given are: A. The z-test using rejection region(s) is used when the population is normal. The z-test using a P-value is used when the population is not normal. B. In the z-test using rejection region(s), the test statistic is compared with critical values. The z-test using a P-value compares the P-value with the level of significance α C. The z-test using rejection region(s) is used when the population is not normal. The z-test using a P-value is used when the population is normal D. In the z-test using rejection region(s), the test statistic is compared with the level of significance α. The z-test using a P-value compares the P-value with the critical values.

Step 3 ：From my understanding of z-tests, the difference between the two methods is not about whether the population is normal or not, but rather about what is being compared (test statistic vs critical values or P-value vs level of significance). Therefore, options A and C can be eliminated.

Step 4 ：Between options B and D, option B seems to correctly describe the process of each method: in the rejection region(s) approach, the test statistic (z-score) is compared with critical values to determine whether to reject the null hypothesis, while in the P-value approach, the P-value is compared with the level of significance to make this determination.

Step 5 ：Therefore, the correct answer is option B.

Step 6 ：\(oxed{\text{B. In the z-test using rejection region(s), the test statistic is compared with critical values. The z-test using a P-value compares the P-value with the level of significance } \alpha}\)