We are tasked with performing a hypothesis test based on the information given:

• Sample mean ($\bar{x}$) = 54
• Sample size ($n$) = 41
• Population standard deviation ($\sigma$) = 5.6
• Null hypothesis ($H_0$): $\mu = 56$
• Alternative hypothesis ($H_a$): $\mu < 56$
• Significance level ($\alpha$) = 0.05

Steps for the hypothesis test:

1. Identify the type of test:

   • Since $H_a$ is $\mu < 56$, this is a left-tailed z-test for the population mean.

2. Calculate the z-test statistic: The z-test statistic is given by:

   $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ Substituting the values:

   $z = \frac{54 - 56}{\frac{5.6}{\sqrt{41}}} = \frac{-2}{\frac{5.6}{6.403}} = \frac{-2}{0.8748} \approx -2.29$

3. Determine the critical value: For a significance level of $\alpha = 0.05$, the critical value for a left-tailed z-test is approximately $-1.645$.

4. Make a decision:

   • If the calculated z-value is less than the critical value, we reject the null hypothesis.
   • In this case, $z = -2.29$ is less than $-1.645$, so we reject the null hypothesis.

Conclusion:

There is sufficient evidence to reject $H_0$, meaning that the sample data supports the claim that $\mu < 56$ at the 0.05 significance level.

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Would you like more details or explanations on any step?

Here are 5 related questions you might want to explore:

1. How is the z-test statistic calculated when the population standard deviation is unknown?
2. What is the difference between a one-tailed and a two-tailed test?
3. What happens if the significance level ($\alpha$) is changed?
4. How do we interpret a Type I error in this hypothesis test?
5. How can we calculate the p-value for this z-test?

Tip: When performing hypothesis tests, always make sure to verify whether the test should be one-tailed or two-tailed to ensure proper interpretation.