how to find test statistic of a two sample t test

To find the test statistic for a two-sample t-test, follow these steps:

### Formula for the test statistic:
For a two-sample t-test, the test statistic (\( t \)) is calculated as:

\[
t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]

Where:
- \( \bar{X}_1 \) = sample mean of group 1
- \( \bar{X}_2 \) = sample mean of group 2
- \( s_1^2 \) = sample variance of group 1
- \( s_2^2 \) = sample variance of group 2
- \( n_1 \) = sample size of group 1
- \( n_2 \) = sample size of group 2

### Steps:
1. **Calculate the sample means**: Find the average for each sample.
   \[
   \bar{X}_1 = \frac{\sum X_1}{n_1}, \quad \bar{X}_2 = \frac{\sum X_2}{n_2}
   \]
   
2. **Calculate the sample variances**: Compute the variance for each sample.
   \[
   s_1^2 = \frac{\sum (X_1 - \bar{X}_1)^2}{n_1 - 1}, \quad s_2^2 = \frac{\sum (X_2 - \bar{X}_2)^2}{n_2 - 1}
   \]
   
3. **Substitute the values** into the formula above to get the test statistic \( t \).

### Assumptions:
- **Equal or unequal variances**: If you assume equal variances, you pool the variances before calculation. Otherwise, use the variances individually as shown.
- **Degrees of freedom (df)**: For unequal variances, degrees of freedom are calculated using the Welch-Satterthwaite equation. For equal variances, use:
   \[
   df = n_1 + n_2 - 2
   \]

Would you like a detailed example or any further clarifications?

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Here are five questions to expand your understanding:
1. How do you calculate pooled variance in a two-sample t-test?
2. What assumptions are necessary for a two-sample t-test?
3. How do you interpret the p-value after calculating the t-statistic?
4. When should you use a two-tailed vs. one-tailed t-test?
5. What is the difference between independent and paired two-sample t-tests?

**Tip**: Always verify if the variances between the two groups are equal before deciding the t-test type.

Learn how to calculate the test statistic for a two-sample t-test with step-by-step instructions. This guide covers the formulas for sample means, variances, and degrees of freedom. Ideal for undergraduate students studying statistics or hypothesis testing.

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