Two Sample t-Test (Pooled-Variance) Calculator

Understanding the Two Sample T-Test (Pooled Variance)

Two Sample T-Test (Pooled Variance) is used to determine whether there is a significant difference between the means of two independent groups, assuming equal variances for both groups. This test combines variances to calculate a pooled standard deviation for a more accurate comparison of group means.

When to Use the Two Sample T-Test (Pooled Variance)

This test is appropriate when the two groups have similar variances and are approximately normally distributed. It provides accurate estimations under these assumptions.

Real-Life Example: Comparing Two Training Programs

Suppose a researcher investigates the effectiveness of two training programs by comparing participants' test scores:

• Group 1: Participants in Program A
• Group 2: Participants in Program B

The collected data shows:

• Mean score for Group 1 = 78, Standard Deviation = 8, Sample Size = 25
• Mean score for Group 2 = 82, Standard Deviation = 9, Sample Size = 30

Using the Two Sample T-Test (Pooled Variance), the researcher can test if Program B significantly differs from Program A in terms of effectiveness. Here are the calculation steps:

Step-by-Step Calculation

1. Calculate the pooled standard deviation:

$$ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} $$

2. Calculate the t-score:

$$ t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} $$

where:

• $ \bar{X}_1 $ and $ \bar{X}_2 $ are the means of the two groups,
• $ s_p $ is the pooled standard deviation,
• $ n_1 $ and $ n_2 $ are the sample sizes of the two groups.

3. Calculate the Degrees of Freedom

In the two-sample t-test with pooled variance, we calculate the degrees of freedom to determine the appropriate t-distribution to compare our t-score against. Since this test assumes equal variances across both groups, we can use a straightforward formula:

$$ df = n_1 + n_2 - 2 $$

This formula reflects the combined sample sizes of both groups, minus 2. By pooling variances, we treat both groups as though they are samples from a single population with a shared variance. This simplifies the degrees of freedom calculation, allowing for a more direct approach compared to tests that assume unequal variances. The result is an efficient way to account for both samples under the equal variance assumption, providing sufficient information to perform an accurate hypothesis test.

Result and Interpretation

The calculated t-score is approximately -1.725, with 53 degrees of freedom. The p-value is approximately 0.0903. Since the p-value is greater than the significance level of 0.05, we do not reject the null hypothesis. This indicates that there is no statistically significant difference between the two programs’ effectiveness at this level.

Hypothesis Testing

This test can be conducted as a right-tailed, left-tailed, or two-tailed test. The hypothesis type determines how we interpret the t-score and p-value:

• Right-tailed: Tests if Group 1's mean is significantly greater than Group 2's. For a right-tailed test, the hypothesis is \( H_0: \bar{X}_1 \leq \bar{X}_2 \) vs. \( H_1: \bar{X}_1 > \bar{X}_2 \). We use the calculated t-score directly, and the p-value is obtained as \( P(T > t) \), where \( T \) follows the t-distribution with \( df \) degrees of freedom.
• Left-tailed: Tests if Group 1's mean is significantly less than Group 2's. For a left-tailed test, the hypothesis is \( H_0: \bar{X}_1 \geq \bar{X}_2 \) vs. \( H_1: \bar{X}_1 < \bar{X}_2 \). We use the negative of the calculated t-score, and the p-value is \( P(T < t) \).
• Two-tailed: Tests if there is any significant difference between the two means. For a two-tailed test, the hypothesis is \( H_0: \bar{X}_1 = \bar{X}_2 \) vs. \( H_1: \bar{X}_1 \neq \bar{X}_2 \). We use the absolute value of the t-score, and the p-value is calculated as \( 2 \times P(T > |t|) \).

In each case, if the p-value is less than the chosen significance level (e.g., 0.05), we reject the null hypothesis in favor of the alternative hypothesis.

How to Obtain the P-Value from the T-Score

Once the t-score is calculated, we need the corresponding p-value to determine the significance of our result. There are two main ways to obtain the p-value:

• • Using Statistical Libraries: Many statistical libraries, such as jStat in JavaScript, scipy.stats in Python, and R, provide functions to calculate the p-value directly from the t-score. For example, in Python, you can use:

from scipy.stats import t
p_value = 2 * (1 - t.cdf(abs(t_score), df))

• Using a T-Distribution Table: If statistical libraries are not available, you can also use a t-distribution table, which provides critical values for various degrees of freedom and significance levels. Match your calculated t-score with the critical value in the table to find the approximate p-value. Note that t-tables are typically available in statistics textbooks and online resources.