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In statistical inference, a common and critical task is determining whether two independent groups have meaningfully different population means. This problem requires implementing a robust statistical test that handles the practical reality of unequal variances between groups—a situation frequently encountered in real-world experimental data.
When comparing means from two independent samples, we must account for the uncertainty inherent in estimating population parameters from sample data. The Welch's approach provides a powerful solution by:
Test Statistic: $$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$
Where $\bar{X}_1$ and $\bar{X}_2$ are sample means, $s_1^2$ and $s_2^2$ are sample variances, and $n_1$ and $n_2$ are sample sizes.
Welch-Satterthwaite Degrees of Freedom: $$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}$$
Cohen's d Effect Size: $$d = \frac{\bar{X}_1 - \bar{X}2}{s{pooled}}$$
Where the pooled standard deviation is: $$s_{pooled} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$$
Cohen's d provides guidelines for interpreting the magnitude of the difference:
Implement a function that performs a complete independent samples mean comparison. Given two datasets and a significance level, your function should return a comprehensive results dictionary containing the test statistic, p-value, degrees of freedom, hypothesis decision, and effect size measure.
sample1 = [12, 14, 13, 15, 14]
sample2 = [8, 9, 10, 9, 11]
alpha = 0.05{'t_statistic': 5.8244, 'p_value': 0.000325, 'degrees_of_freedom': 8.0, 'reject_null': True, 'cohens_d': 3.6836}Step-by-step calculation:
Sample Statistics:
Standard Error:
Test Statistic:
Degrees of Freedom:
P-value:
Decision:
Effect Size:
The extremely large effect size and tiny p-value indicate a highly significant and practically meaningful difference between the two groups.
sample1 = [10, 11, 12, 13, 14]
sample2 = [20, 21, 22, 23, 24]
alpha = 0.05{'t_statistic': -10.0, 'p_value': 8e-06, 'degrees_of_freedom': 8.0, 'reject_null': True, 'cohens_d': -6.3246}Analysis:
Sample Statistics:
Difference: The means differ by exactly 10 units.
Test Statistic:
P-value:
Effect Size:
This represents a textbook case of strongly separated groups with virtually no overlap between distributions.
sample1 = [50, 52, 48, 51, 49]
sample2 = [51, 49, 50, 52, 48]
alpha = 0.05{'t_statistic': 0.0, 'p_value': 1.0, 'degrees_of_freedom': 8.0, 'reject_null': False, 'cohens_d': 0.0}Analysis of identical distributions:
Sample Statistics:
Test Statistic:
P-value:
Decision:
Effect Size:
This example demonstrates the scenario where there is genuinely no difference between groups. The statistical test correctly identifies that any observed differences are purely due to random variation.
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