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.