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.