We are given the following information for the hypothesis test:

• Null hypothesis ($H_0$): $\mu \leq 45$
• Alternative hypothesis ($H_A$): $\mu > 45$
• Significance level ($\alpha = 0.02$)
• Sample size ($n = 80$)
• Population standard deviation ($\sigma = 9$)
• Sample mean ($\bar{x} = 47.1$)

To find the $p$-value for this test, we need to use the Z-test for the population mean, since the population standard deviation ($\sigma$) is known.

Step 1: Calculate the Z-score

The formula for the Z-score is:

$Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $\bar{x} = 47.1$ (sample mean)
• $\mu_0 = 45$ (the hypothesized population mean under $H_0$)
• $\sigma = 9$ (population standard deviation)
• $n = 80$ (sample size)

Substitute the values into the formula:

$Z = \frac{47.1 - 45}{\frac{9}{\sqrt{80}}}$

First, calculate the standard error:

$\text{Standard Error} = \frac{9}{\sqrt{80}} = \frac{9}{8.944} \approx 1.006$

Now, calculate the Z-score:

$Z = \frac{47.1 - 45}{1.006} = \frac{2.1}{1.006} \approx 2.088$

Step 2: Find the p-value

Since this is a one-tailed test (the alternative hypothesis is $\mu > 45$), we need to find the area to the right of the Z-score.

Using the standard normal distribution table or a calculator, we find that the probability corresponding to $Z = 2.088$ is approximately $0.9817$. The area to the right is:

$p\text{-value} = 1 - 0.9817 = 0.0183$

Step 3: Conclusion

The $p$-value is approximately 0.0183, which corresponds to option C.

Answer:

C. 0.0183

Would you like further details on any of the steps? Here are 5 related questions you might want to explore:

1. What are the assumptions behind using a Z-test for population means?
2. How do you interpret the p-value in hypothesis testing?
3. What happens if the significance level $\alpha$ is larger than the p-value?
4. What is the difference between one-tailed and two-tailed tests?
5. How does increasing the sample size affect the Z-score?

Tip: When conducting hypothesis tests, always check the conditions for the test you're using (e.g., normality, known variance) to ensure valid results.