Questions: Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ=40 ; α=0.09 ; σ=3.09. Sample statistics: x̄=39.6, n=50 Identify the null and alternative hypotheses. Choose the correct answer below. A. H₀: μ<40 Hₐ: μ=40 B. H₀: μ>40 C. H₀: p ≠ 40 Hₐ: μ=40 D. H₀: μ=40 Hₐ: μ ≠ 40 E. H₀: μ=40 F. H₀: μ=40 Hₐ: μ<40 Calculate the standardized test statistic. The standardized test statistic is -0.92. (Round to two decimal places as needed.) Determine the critical value(s). Select the correct choice below and fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The critical values are ±0.92. B. The critical value is .

Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ=40 ; α=0.09 ; σ=3.09. Sample statistics: x̄=39.6, n=50 Identify the null and alternative hypotheses. Choose the correct answer below. A. H₀: μ<40 Hₐ: μ=40 B. H₀: μ>40 C. H₀: p ≠ 40 Hₐ: μ=40 D. H₀: μ=40 Hₐ: μ ≠ 40 E. H₀: μ=40 F. H₀: μ=40 Hₐ: μ<40 Calculate the standardized test statistic. The standardized test statistic is -0.92. (Round to two decimal places as needed.) Determine the critical value(s). Select the correct choice below and fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The critical values are ±0.92. B. The critical value is .
Transcript text: Test the claim about the population mean, $\mu$, at the given level of significance using the given sample statistics. Claim: $\mu=40 ; \alpha=0.09 ; \sigma=3.09$. Sample statistics: $\bar{x}=39.6, n=50$ Identify the null and alternative hypotheses. Choose the correct answer below. A. $\mathrm{H}_{0}: \mu<40$ $H_{a}: \mu=40$ B. $H_{0}: \mu>40$ C. $H_{0}: p \neq 40$ $H_{\mathrm{a}}: \mu=40$ D. \[ \begin{array}{l} H_{0}: \mu=40 \\ H_{a}: \mu \neq 40 \end{array} \] (E) $\mathrm{H}_{0}: \mu=40$ F. $H_{0}: \mu=40$ $H_{a}: \mu<40$ Calculate the standardized test statistic. The standardized test statistic is -0.92 . (Round to two decimal places as needed.) Determine the critical value(s). Select the correct choice below and fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The critical values are $\pm 0.92$. B. The critical value is .
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Solution

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Solution Steps

Step 1: Standard Error Calculation

The standard error \( SE \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3.09}{\sqrt{50}} \approx 0.44 \]

Step 2: Test Statistic Calculation

The test statistic \( Z \) is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{39.6 - 40}{0.44} \approx -0.92 \]

Step 3: P-value Calculation

For a two-tailed test, the p-value \( P \) is calculated as: \[ P = 2 \times (1 - T(|z|)) \approx 0.36 \]

Step 4: Critical Values Determination

For a significance level \( \alpha = 0.09 \) in a two-tailed test, the critical values are: \[ \text{Critical Values} = -1.7 \text{ and } 1.7 \]

Final Answer

The test statistic is \( Z \approx -0.92 \), the p-value is \( P \approx 0.36 \), and the critical values are \( -1.7 \) and \( 1.7 \).

Thus, the final boxed answer is: \[ \boxed{Z \approx -0.92, \; P \approx 0.36, \; \text{Critical Values: } -1.7 \text{ and } 1.7} \]

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