Questions: Explain the difference between the z-test for μ using rejection region(s) and the z-test for μ using a P-value. Choose the correct answer below. A. 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. B. 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. C. 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. D. 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.

Explain the difference between the z-test for μ using rejection region(s) and the z-test for μ using a P-value.

Choose the correct answer below.
A. 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.
B. 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.
C. 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.
D. 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.

Transcript text: 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 not normal. The $z$-test using a P -value is used when the population is normal. B. 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. C. 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 $\alpha$. D. 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.

Solution

Solution Steps

Step 1: Calculate the Z-score

The Z-score is calculated using the formula:

\[ z = \frac{X - \mu}{\sigma} \]

Substituting the values:

\[ z = \frac{1.96 - 0}{1} = 1.96 \]

Thus, the Z-score is $ z = 1.96 $.

Step 2: Calculate the Standard Error

The Standard Error $ SE $ is calculated as follows:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{30}} \approx 2.7386 \]

Step 3: Calculate the Test Statistic

The test statistic $ Z_{test} $ is calculated using the formula:

\[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} \]

Substituting the values:

\[ Z_{test} = \frac{105 - 100}{2.7386} \approx 1.8257 \]

Step 4: Calculate the P-value

For a two-tailed test, the P-value is calculated as:

\[ P = 2 \times (1 - T(|z|)) \approx 0.0679 \]

Step 5: Compare the Results

In the z-test using rejection region(s), the test statistic $ Z_{test} $ is compared with critical values. In contrast, the z-test using a P-value compares the P-value with the level of significance $ \alpha $.

Final Answer

The correct answer is:

$\boxed{C}$

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