Hypothesis Testing Using T-Test: Complete Guide

Quick answer: Learn the t-test for hypothesis testing, including one-sample, two-sample and paired t-tests, when to use each, and how to interpret results in Python.

What is a T-Test?

A t-test is a statistical hypothesis test used to determine whether the means of one or two groups differ significantly, when the population standard deviation is unknown and must be estimated from the sample itself. This is the most commonly used hypothesis test in practical data analysis.

Three Types of T-Tests

One-Sample T-Test Example

from scipy import stats

scores = [78, 82, 85, 90, 76, 88, 79, 84]
hypothesized_mean = 80

t_stat, p_value = stats.ttest_1samp(scores, hypothesized_mean)
print(f"t-statistic: {t_stat:.2f}, p-value: {p_value:.4f}")

Independent Two-Sample T-Test Example

group_a = [78, 82, 85, 90, 76]
group_b = [65, 70, 72, 68, 74]

t_stat, p_value = stats.ttest_ind(group_a, group_b)
print(f"t-statistic: {t_stat:.2f}, p-value: {p_value:.4f}")

Paired T-Test Example

before = [70, 75, 68, 80, 72]
after  = [78, 82, 74, 85, 79]

t_stat, p_value = stats.ttest_rel(before, after)
print(f"t-statistic: {t_stat:.2f}, p-value: {p_value:.4f}")

A paired test is used here because each before and after value belongs to the same subject, which the independent two-sample test would incorrectly ignore.

Interpreting the P-Value

If the p-value falls below the chosen significance level, typically 0.05, the difference is considered statistically significant, meaning it is unlikely to be due to random chance alone under the null hypothesis of no real difference.

Assumptions of the T-Test

• Data is approximately normally distributed, particularly important for small sample sizes

• Observations are independent (except in the paired test, which specifically handles related measurements)

• For the independent two-sample test, variances between the two groups are roughly equal, unless using Welch's correction, which relaxes this assumption

T-Test vs ANOVA

A t-test compares exactly two groups. When comparing three or more groups, ANOVA is the correct choice instead, since running multiple t-tests would inflate the overall false positive rate.

Common Interview Questions

When would you use a paired t-test instead of an independent two-sample t-test?

When the two sets of measurements come from the same subjects, such as before and after a treatment, since the paired test accounts for the natural correlation between each subject's own two measurements.

What do you do if the equal variance assumption is violated in a two-sample t-test?

Use Welch's t-test instead, a variant that does not assume equal variances between the two groups, available in most statistical software as an alternative option.