Questions: Use the provided information to test the following claim. H0: μ=12 Ha: μ>12 X̄=13, σ=4 n=49, α=0.01 (a) Rejection Region Method (b) P-Value Method

Solution Steps

To test the claim about the population mean using the provided information, we can use two methods: the Rejection Region Method and the P-Value Method.

(a) Rejection Region Method:

1. Calculate the test statistic using the formula for the z-test: \( z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \).
2. Determine the critical value for \( \alpha = 0.01 \) from the standard normal distribution table.
3. Compare the test statistic to the critical value to decide whether to reject the null hypothesis.

(b) P-Value Method:

1. Calculate the test statistic as in the Rejection Region Method.
2. Find the p-value corresponding to the test statistic using the standard normal distribution.
3. Compare the p-value to \( \alpha = 0.01 \) to decide whether to reject the null hypothesis.

The test statistic \( z \) is calculated using the formula:

\[ z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}} = \frac{13 - 12}{4 / \sqrt{49}} = \frac{1}{\frac{4}{7}} = 1.75 \]

For a significance level of \( \alpha = 0.01 \) in a one-tailed test, the critical value from the standard normal distribution is:

\[ z_{\text{critical}} = 2.3263 \]

We compare the test statistic \( z \) with the critical value:

\[ 1.75 < 2.3263 \]

Since the test statistic does not exceed the critical value, we do not reject the null hypothesis \( H_0 \).

Next, we calculate the p-value associated with the test statistic:

\[ p\text{-value} = 1 - \Phi(z) = 1 - \Phi(1.75) \approx 0.0401 \]

We compare the p-value with the significance level \( \alpha \):

\[ 0.0401 > 0.01 \]

Since the p-value is greater than \( \alpha \), we do not reject the null hypothesis \( H_0 \).

Final Answer