Questions: Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ=50 ; α=0.02 ; σ=3.52. Sample statistics: x̄=49.3, n=56 Identify the null and alternative hypotheses. Choose the correct answer below. A. H0: μ=50 B. H0: μ=50 Ha: μ<50 Ha: μ>50 C. H0: μ>50 D. H0: μ<50 Ha: μ=50 Ha: μ=50 E. H0: μ=50 F. H0: μ ≠ 50 Ha: μ ≠ 50 Ha: μ=50

Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ=50 ; α=0.02 ; σ=3.52. Sample statistics: x̄=49.3, n=56

Identify the null and alternative hypotheses. Choose the correct answer below.
A. H0: μ=50 B. H0: μ=50 Ha: μ<50 Ha: μ>50
C. H0: μ>50 D. H0: μ<50 Ha: μ=50 Ha: μ=50
E. H0: μ=50 F. H0: μ ≠ 50 Ha: μ ≠ 50 Ha: μ=50

Transcript text: Test the claim about the population mean, $\mu$, at the given level of significance using the given sample statistics. Claim: $\mu=50 ; \alpha=0.02 ; \sigma=3.52$. Sample statistics: $\bar{x}=49.3, n=56$ Identify the null and alternative hypotheses. Choose the correct answer below. A. $\mathrm{H}_{0}: \mu=50$ B. $H_{0}: \mu=50$ $H_{a}: \mu<50$ $\mathrm{H}_{\mathrm{a}}: \mu>50$ C. $\mathrm{H}_{0}: \mu>50$ D. $H_{0}: \mu<50$ $H_{a}: \mu=50$ $\mathrm{H}_{\mathrm{a}}: \mu=50$ E. $H_{0}: \mu=50$ F. $H_{0}: \mu \neq 50$ $\mathrm{H}_{\mathrm{a}}: \mu \neq 50$ $\mathrm{H}_{\mathrm{a}}: \mu=50$

Solution

Solution Steps

Step 1: Calculate the Standard Error

The standard error $ SE $ is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3.52}{\sqrt{56}} \approx 0.4704 \]

Step 2: Calculate the Test Statistic

The test statistic $ Z $ is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{49.3 - 50}{0.4704} \approx -1.4882 \]

Step 3: Calculate the P-value

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

\[ P = 2 \times (1 - T(|z|)) \approx 0.1367 \]

Step 4: State the Hypotheses

The null and alternative hypotheses are defined as follows:

\[ H_0: \mu = 50 \] \[ H_a: \mu \neq 50 \]

Final Answer

Based on the calculations, we conclude that the test statistic is $ Z \approx -1.4882 $ and the P-value is $ 0.1367 $. Since the P-value $ 0.1367 $ is greater than the significance level $ \alpha = 0.02 $, we fail to reject the null hypothesis.

Thus, the answer is:

\[ \boxed{H_0: \mu = 50} \]

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