Use s to calculate the value of the test statistic.

To calculate the value of the test statistic using \( s \), which is often the sample standard deviation, we need more information about the context of the problem. The most common formulae involve comparing sample means to population means or differences between means, for instance:

### Common test statistic formulae:

1. **For a single-sample t-test**:
   \[
   t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
   \]
   - \(\bar{x}\): sample mean
   - \(\mu\): population mean
   - \(s\): sample standard deviation
   - \(n\): sample size

2. **For a two-sample t-test**:
   \[
   t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
   \]
   - \(\bar{x}_1, \bar{x}_2\): means of the two samples
   - \(s_p\): pooled standard deviation
   - \(n_1, n_2\): sizes of the two samples
   - The pooled standard deviation (\(s_p\)) is calculated as:
     \[
     s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}
     \]

3. **For a z-test (if sample size is large or population standard deviation is known)**:
   \[
   z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
   \]

The choice of formula depends on whether we are dealing with one sample or two, whether the population standard deviation is known, and whether the sample size is large enough to use the normal distribution instead of the t-distribution.

Could you provide more details, such as the type of test (one-sample, two-sample, etc.), sample mean (\(\bar{x}\)), population mean (\(\mu\)), and the sample size (\(n\))? That way, I can calculate the test statistic precisely.

### Related Questions:
1. What is the difference between using \( s \) (sample standard deviation) and \( \sigma \) (population standard deviation) in a test statistic?
2. How does sample size \( n \) affect the value of the test statistic?
3. When should you use a t-test versus a z-test?
4. How do you interpret a p-value in relation to the test statistic?
5. What assumptions need to be met for a two-sample t-test?

**Tip:** Always check if the sample size is large enough (\( n > 30 \)) to use the z-test approximation instead of the t-test.

Learn how to calculate the value of the test statistic using the sample standard deviation (s) with examples of t-tests and z-tests. Includes key formulas for hypothesis testing, such as the single-sample t-test and the two-sample t-test, with a breakdown of when to use each. Suitable for college-level statistics students.

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