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To solve this hypothesis test problem, let's go through each part step-by-step:

### Given Data
- **Null Hypothesis (\(H_0\))**: \(\mu = 74.7\)
- **Alternative Hypothesis (\(H_A\))**: \(\mu \neq 74.7\)
- **Significance Level** (\(\alpha\)): 0.01
- Sample size (\(n\)): 28
- Sample mean (\(\bar{x}\)): 79.1
- Sample standard deviation (\(s\)): 9.8

### Step 1: Calculate the Test Statistic
Since we do not know the population standard deviation, we will use the t-test statistic for a sample mean.

The t-test statistic is calculated by:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]

Substituting the values:
\[
t = \frac{79.1 - 74.7}{\frac{9.8}{\sqrt{28}}}
\]

1. Calculate the denominator \(\frac{9.8}{\sqrt{28}}\):
   - \(\sqrt{28} \approx 5.2915\)
   - \(\frac{9.8}{5.2915} \approx 1.8526\)

2. Now calculate the test statistic:
   \[
   t = \frac{79.1 - 74.7}{1.8526} \approx \frac{4.4}{1.8526} \approx 2.375
   \]

Thus, the **test statistic** is approximately **2.375** (rounded to three decimal places).

### Step 2: Calculate the P-value
Since this is a two-tailed test, we need to find the p-value associated with \( t = 2.375 \) for a t-distribution with \( n - 1 = 27 \) degrees of freedom.

Using a t-table or statistical software, we find the two-tailed p-value for \( t = 2.375 \) with 27 degrees of freedom.

For approximate results:
- A t-score of 2.375 with 27 degrees of freedom yields a two-tailed p-value around **0.0254** (accurate to four decimal places).

### Step 3: Decision Rule and Conclusion
- Compare the p-value (0.0254) with the significance level (\(\alpha = 0.01\)).
- Since \(0.0254 > 0.01\), the p-value is **greater than \(\alpha\)**.

**Conclusion**:
- We **fail to reject the null hypothesis** because the p-value is greater than the significance level.
- There is **not sufficient evidence to support the claim that the population mean is not equal to 74.7** at the 0.01 significance level.

---

Would you like further details on any of these steps or have additional questions?

### Related Questions:
1. How would the conclusion change if the significance level was set to 0.05 instead of 0.01?
2. What is the effect of increasing the sample size on the test statistic?
3. How would this test change if we were testing a one-tailed hypothesis?
4. What assumptions are we making about the population distribution for this test?
5. How can we interpret the meaning of the p-value in practical terms?

#### Tip:
When conducting a hypothesis test, always ensure the sample size and standard deviation are accurately used, especially if you don't know the population standard deviation, as this affects the choice between using a t-test or z-test.

You wish to test the following claim (H1) at a significance level of α = 0.01. H0: μ = 74.7, HA: μ ≠ 74.7. You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 28 with mean x̄ = 79.1 and a standard deviation of s = 9.8. What is the test statistic for this sample? What is the p-value for this sample? The p-value is less than or equal to α or greater than α. This test statistic leads to a decision to reject, accept, or fail to reject the null. Final conclusion based on the result.

Learn how to perform a two-tailed hypothesis test on a sample mean with unknown population standard deviation at a 0.01 significance level. This example walks through calculating the t-test statistic and finding the p-value for a sample of 28 with a mean of 79.1 and a standard deviation of 9.8.

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