For the following hypothesis​ test:
Start 3 By 1 Matrix 1st Row 1st Column Upper H 0 : mu less than or equals 45 2nd Row 1st Column Upper H Subscript Upper A Baseline : mu greater than 45 3rd Row 1st Column alpha equals 0.02 EndMatrix H0 : μ≤45	HA : μ > 45	       α=0.02
With nequals=​80, sigmaσequals=​9, and x overbarxequals=​47.1, state the appropriate​ p-value.
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Part 1
A.
0.0314
B.
0.0218
C.
0.0183
D.
0.0512

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.

For the following hypothesis test: H0 : μ≤45, HA : μ > 45, α=0.02. With n=80, σ=9, and x̄=47.1, state the appropriate​ p-value.
A. 0.0314
B. 0.0218
C. 0.0183
D. 0.0512

Learn how to conduct a hypothesis test for population mean using a Z-test. Given a sample size of 80, population standard deviation of 9, and sample mean of 47.1, determine the p-value to test if μ > 45 at a significance level of 0.02. This solution is suitable for undergraduate statistics students.

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