how do i find test statistic on the average number of two samples on a two sample t test

To find the test statistic for a two-sample **t-test** on the average (mean) of two samples, follow these steps:

### 1. **Determine the Hypotheses**:
- **Null Hypothesis (H₀)**: There is no difference between the two population means (e.g., \( \mu_1 = \mu_2 \)).
- **Alternative Hypothesis (H₁)**: There is a difference between the two population means (e.g., \( \mu_1 \neq \mu_2 \), or one is greater than the other).

### 2. **Formula for the Test Statistic**:
The test statistic \( t \) for comparing the means of two independent samples is:

\[
t = \frac{ \overline{X_1} - \overline{X_2} }{ \sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} } }
\]

Where:
- \( \overline{X_1} \) and \( \overline{X_2} \) are the sample means for group 1 and group 2, respectively.
- \( s_1^2 \) and \( s_2^2 \) are the sample variances (square of the standard deviations) of the two samples.
- \( n_1 \) and \( n_2 \) are the sample sizes for group 1 and group 2, respectively.

### 3. **Steps to Compute**:
- **Step 1**: Calculate the mean \( \overline{X_1} \) and \( \overline{X_2} \) for each sample.
- **Step 2**: Compute the sample variances \( s_1^2 \) and \( s_2^2 \).
- **Step 3**: Use the formula to calculate the test statistic \( t \).

### 4. **Degrees of Freedom (df)**:
For unequal variances (Welch’s t-test), use:

\[
df = \frac{ \left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2 }{ \frac{ \left( \frac{s_1^2}{n_1} \right)^2 }{n_1 - 1} + \frac{ \left( \frac{s_2^2}{n_2} \right)^2 }{n_2 - 1} }
\]

For equal variances, the degrees of freedom are \( df = n_1 + n_2 - 2 \).

### 5. **Interpret the Test Statistic**:
Once you have calculated the \( t \)-statistic, compare it to the critical \( t \)-value from the t-distribution table with the appropriate degrees of freedom and significance level (α, typically 0.05). This will tell you whether to reject or fail to reject the null hypothesis.

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Would you like more detailed steps on calculating variances or comparing \( t \)-values? Here are five related questions to deepen your understanding:

1. How do you calculate pooled variance for a two-sample t-test with equal variances?
2. What is the difference between a one-tailed and two-tailed t-test?
3. How do you interpret the p-value in the context of a two-sample t-test?
4. What assumptions must be satisfied for a two-sample t-test to be valid?
5. Can the two-sample t-test be used for paired data, and how would the method change?

### Tip:
Ensure the sample sizes are sufficiently large or check for normality in your data, as t-tests are sensitive to non-normal distributions, especially with small samples.

Learn how to calculate the test statistic for a two-sample t-test to compare the average values of two independent samples. This guide covers the key formulas, including the test statistic formula and degrees of freedom for equal and unequal variances.

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