T Test

Introduction

Purpose: test hypotheses about population means using sample data. Basis: Student’s t distribution adjusts for small samples and unknown population variance. Application: determine if sample mean differs from known value or between groups. Key concept: compare calculated t statistic to critical values for significance.

"The t test is the most important tool for comparing means when population variance is unknown." -- William Sealy Gosset

History and Origin

Development

Invented by William Sealy Gosset in 1908 under pseudonym “Student” at Guinness Brewery to analyze small sample data in quality control.

Publication

Published in 1908 in Biometrika; introduced Student’s t distribution as alternative to normal distribution for small samples.

Impact

Foundation of modern hypothesis testing; widely adopted in statistics, medicine, social sciences.

Types of T Tests

One-Sample T Test

Tests if sample mean differs from known or hypothesized population mean.

Independent Samples T Test

Compares means of two independent groups to assess difference.

Paired Samples T Test

Compares means of two related groups, e.g., before-after measurements.

One-Tailed vs Two-Tailed

One-tailed tests directional hypotheses; two-tailed test non-directional.

Assumptions

Normality

Data in each group approximately normally distributed. Robust for moderate deviations if sample size > 30.

Independence

Observations independent within and between groups.

Scale of Measurement

Dependent variable measured at interval or ratio scale.

Homogeneity of Variance

For independent samples t test: equal variances assumed or addressed by Welch’s correction.

Formulas and Calculations

One-Sample T Test

t = (X̄ - μ₀) / (s / √n)Where: X̄ = sample mean μ₀ = hypothesized population mean s = sample standard deviation n = sample size

Independent Samples T Test (Equal Variances)

t = (X̄₁ - X̄₂) / (s_p * √(1/n₁ + 1/n₂))Where: X̄₁, X̄₂ = sample means n₁, n₂ = sample sizes s_p = pooled standard deviation s_p = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁ + n₂ - 2)]

Paired Samples T Test

t = D̄ / (s_D / √n)Where: D̄ = mean of differences s_D = standard deviation of differences n = number of pairs

Degrees of Freedom

One-sample and paired: df = n - 1; independent samples (equal variances): df = n₁ + n₂ - 2; Welch’s t test uses adjusted df.

Testing Procedure

Step 1: State Hypotheses

Null hypothesis (H₀): no difference or effect. Alternative hypothesis (H₁): difference exists.

Step 2: Choose Significance Level

Common α = 0.05; defines Type I error threshold.

Step 3: Calculate Test Statistic

Compute t value using appropriate formula.

Step 4: Determine Critical Value or p-value

Use t distribution tables or software based on df and α.

Step 5: Decision

Reject H₀ if |t| > critical value or p < α; otherwise do not reject.

Interpretation of Results

Significance

Statistically significant t indicates evidence against H₀; supports alternative hypothesis.

Effect Size

Measure magnitude of difference: Cohen’s d, Glass’s Δ.

Confidence Intervals

Range of plausible values for mean difference computed from t statistic and standard error.

Type I and II Errors

Type I: false positive; Type II: false negative; balancing via α and sample size.

Examples

One-Sample T Test Example

Sample of 15 students: mean IQ = 105, s = 10; test if mean differs from population mean 100 at α=0.05.

Independent Samples T Test Example

Compare test scores of two classes (n₁=20, n₂=22) with means 78 and 83, s₁=8, s₂=7.

Paired Samples T Test Example

Measure weight of 10 subjects before and after diet; test mean difference.

Advantages and Limitations

Advantages

Simple to compute; applicable to small samples; robust to moderate normality violations; interpretable results.

Limitations

Assumes normality; sensitive to outliers; limited to mean comparisons; requires independent observations; less powerful with unequal variances if uncorrected.

Alternatives

Mann-Whitney U test for nonparametric independent samples; Wilcoxon signed-rank for paired data; ANOVA for multiple groups.

Relationship with Other Tests

ANOVA

Generalizes t test to more than two groups; t test is special case of ANOVA with two groups.

Z Test

Requires known population variance and large samples; t test preferred when variance unknown or sample small.

Nonparametric Tests

Used when t test assumptions violated; rely on rank-based methods.

Software Implementation

SPSS

Menus: Analyze > Compare Means > Independent-Samples T Test or Paired-Samples T Test; outputs include t value, df, p-value.

R

Functions: t.test(x, mu=, alternative=, paired=, var.equal=); returns statistic, p-value, confidence interval.

Python

Libraries: SciPy stats.ttest_1samp(), stats.ttest_ind(), stats.ttest_rel(); detailed documentation available.

References

• Gosset, W.S., "The probable error of a mean," Biometrika, vol. 6, no. 1, 1908, pp. 1-25.
• Student, "The probable error of a mean," Biometrika, vol. 6, 1908, pp. 1-25.
• Student, "On the error of counting with a finite number of observations," Biometrika, vol. 6, 1908, pp. 1-15.
• Ruxton, G.D., "The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test," Behavioral Ecology, vol. 17, no. 4, 2006, pp. 688-690.
• Student, W.S., "Some recent contributions to the theory of testing hypotheses," Journal of the Royal Statistical Society, Series A, vol. 72, 1909, pp. 1-15.