Questions: Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ=40 ; α=0.06 ; σ=3.96. Sample statistics: x̄=39.4, n=54 Identify the null and alternative hypotheses. Choose the correct answer below. A. H₀: μ>40 B. H₀: μ=40 Ha: μ=40 Ha: μ ≠ 40 C. H₀: μ=40 D. H₀: μ=40 Ha: μ>40 Ha: μ<40 E. H₀: μ<40 F. H₀: μ ≠ 40 Ha: μ=40 Ha: μ=40

Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ=40 ; α=0.06 ; σ=3.96. Sample statistics: x̄=39.4, n=54 Identify the null and alternative hypotheses. Choose the correct answer below. A. H₀: μ>40 B. H₀: μ=40 Ha: μ=40 Ha: μ ≠ 40 C. H₀: μ=40 D. H₀: μ=40 Ha: μ>40 Ha: μ<40 E. H₀: μ<40 F. H₀: μ ≠ 40 Ha: μ=40 Ha: μ=40
Transcript text: Test the claim about the population mean, $\mu$, at the given level of significance using the given sample statistics. Claim: $\mu=40 ; \alpha=0.06 ; \sigma=3.96$. Sample statistics: $\bar{x}=39.4, n=54$ Identify the null and alternative hypotheses. Choose the correct answer below. A. $\mathrm{H}_{0}: \mu>40$ B. $H_{0}: \mu=40$ $\mathrm{H}_{\mathrm{a}}: \mu=40$ $H_{a}: \mu \neq 40$ C. $\mathrm{H}_{0}: \mu=40$ D. $H_{0}: \mu=40$ $\mathrm{H}_{\mathrm{a}}: \mu>40$ $H_{a}: \mu<40$ E. $H_{0}: \mu<40$ F. $H_{0}: \mu \neq 40$ $H_{a}: \mu=40$ $\mathrm{H}_{\mathrm{a}}: \mu=40$
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Solution

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Solution Steps

Step 1: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3.96}{\sqrt{54}} \approx 0.5389 \]

Step 2: Calculate the Test Statistic

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

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{39.4 - 40}{0.5389} \approx -1.1134 \]

Step 3: Calculate the P-value

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

\[ P = 2 \times (1 - T(|z|)) \approx 0.2655 \]

Step 4: State the Hypotheses

The null and alternative hypotheses are stated as follows:

\[ H_0: \mu = 40 \] \[ H_a: \mu \neq 40 \]

Step 5: Decision Rule

We compare the P-value to the significance level \( \alpha = 0.06 \):

Since \( P \approx 0.2655 > 0.06 \), we fail to reject the null hypothesis.

Final Answer

The conclusion is that we do not have sufficient evidence to reject the claim that the population mean \( \mu \) is equal to 40.

Thus, the answer is \\(\boxed{H_0: \mu = 40}\\).

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