To recalculate the test statistic, let's follow these steps carefully:

1. **State the Hypotheses**:
   - Null Hypothesis (\( H_0 \)): Typically, \( \mu = \mu_0 \) (mean is equal to a specific value).
   - Alternative Hypothesis (\( H_a \)): \( \mu \neq \mu_0 \) or as specified in the problem.

2. **Identify the Given Information**:
   - Sample Mean (\( \bar{x} \))
   - Population Mean under \( H_0 \) (\( \mu_0 \))
   - Sample Standard Deviation (\( s \))
   - Sample Size (\( n \))

3. **Calculate the Test Statistic**:
   The formula for the test statistic \( t \) in a one-sample \( t \)-test is:
   \[
   t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
   \]

   Substitute the values provided into this formula.

If you provide me with the specific values for the sample mean, hypothesized population mean, sample standard deviation, and sample size, I can complete the calculation for you. 

Would you like to continue with this explanation? Let me know the values, and we’ll proceed. 

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1. What values do I need for calculating a two-sample test statistic?
2. Can the same formula apply to large and small sample sizes?
3. How do critical values relate to the test statistic?
4. How does this differ for a two-tailed vs. a one-tailed test?
5. When is the \( z \)-test preferred over the \( t \)-test?

**Tip:** For small sample sizes (usually \( n < 30 \)), ensure you use the \( t \)-distribution over the \( z \)-distribution.

Learn how to recalculate the test statistic for a hypothesis test with step-by-step guidance. This solution covers key statistical concepts including hypothesis testing, the t-test formula, and using sample data to verify population means. Ideal for college-level statistics students.

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