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Let’s solve the problem step by step:

### Problem Summary
We are conducting a one-tailed hypothesis test with the following parameters:
- Null hypothesis: \( H_0: \mu = 68.5 \)
- Alternative hypothesis: \( H_a: \mu > 68.5 \)
- Significance level: \( \alpha = 0.002 \)
- Sample size: \( n = 20 \)
- Sample mean: \( \bar{x} = 72.3 \)
- Sample standard deviation: \( s = 7.1 \)

### Step 1: Compute the Test Statistic
The test statistic for a t-test is calculated as:

\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\]

Substitute the given values:
- \( \bar{x} = 72.3 \)
- \( \mu_0 = 68.5 \)
- \( s = 7.1 \)
- \( n = 20 \)

\[
t = \frac{72.3 - 68.5}{7.1 / \sqrt{20}} = \frac{3.8}{7.1 / 4.472} = \frac{3.8}{1.588} \approx 2.394
\]

### Step 2: Determine the Degrees of Freedom
Degrees of freedom (\( df \)) is:
\[
df = n - 1 = 20 - 1 = 19
\]

### Step 3: Find the P-Value
Using the t-distribution table or a calculator for a one-tailed test with \( t = 2.394 \) and \( df = 19 \):
- The corresponding \( p \)-value is approximately \( 0.0133 \).

### Step 4: Compare \( P \)-Value with \( \alpha \)
- \( p = 0.0133 \)
- \( \alpha = 0.002 \)

Since \( p > \alpha \), we fail to reject the null hypothesis.

### Step 5: State the Conclusion
- The p-value is **greater than \( \alpha \)**.
- Decision: **Fail to reject the null hypothesis**.
- Final conclusion: **There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 68.5.**

Would you like further clarification or a breakdown of any steps? 

Here are 5 related questions:
1. How do you interpret a p-value in hypothesis testing?
2. What is the role of the significance level (\( \alpha \)) in decision-making?
3. How does sample size affect the t-test calculation?
4. When would you use a one-tailed versus a two-tailed test?
5. How does the t-distribution differ from the normal distribution?

**Tip**: Always ensure your sample size is sufficiently large for more reliable results in hypothesis testing!

You wish to test the following claim (H_a) at a significance level of α = 0.002:
H₀: μ = 68.5
Hₐ: μ > 68.5
You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 20 with mean M = 72.3 and a standard deviation of SD = 7.1. What is the p-value for this sample? Does it lead to rejecting the null hypothesis?

Learn how to perform a one-tailed t-test to evaluate the claim that the population mean exceeds 68.5 at a significance level of 0.002. This example includes calculating the t-statistic, p-value, and making statistical decisions based on the data.

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