# Lecture 15: One-Sample and Two-Sample t-Tests

## Statistics / Biostatistics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Perform and interpret a one-sample t-test
2. Perform and interpret an independent two-sample t-test (pooled and Welch)
3. Perform and interpret a paired t-test
4. Verify the assumptions of each test and identify when violations are problematic
5. Report t-test results in a clinical research format

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## Lecture Content

### I. One-Sample t-Test

The one-sample t-test is used to test whether the population mean differs from a specified value mu_0. The hypotheses are H0: mu = mu_0 versus H1: mu does not equal mu_0 (two-sided), or a one-sided alternative. The test statistic is t = (x-bar - mu_0) / (s / sqrt(n)), with df = n - 1. The assumptions are that the data are a simple random sample and that the population is approximately normally distributed (or n is sufficiently large for the CLT to apply).

As an example, a laboratory claims its mean hemoglobin measurement is 14.0 g/dL. A sample of 20 measurements yields x-bar = 14.8 and s = 1.5. The test statistic is t = (14.8 - 14.0) / (1.5 / sqrt(20)) = 0.8 / 0.335 = 2.385. With df = 19 and a critical value of t_(0.025, 19) = 2.093, since 2.385 > 2.093, we reject H0 and conclude that the mean is significantly different from 14.0.

### II. Independent Two-Sample t-Test

The independent two-sample t-test compares the means of two independent groups. The hypotheses are H0: mu_1 = mu_2 (or equivalently, mu_1 - mu_2 = 0) versus H1: mu_1 does not equal mu_2. Two versions exist.

#### Pooled t-Test

The pooled (Student's) t-test assumes equal population variances. The pooled variance is s_p^2 = [(n1-1)*s1^2 + (n2-1)*s2^2] / (n1 + n2 - 2), and the test statistic is t = (x-bar1 - x-bar2) / [s_p * sqrt(1/n1 + 1/n2)], with df = n1 + n2 - 2. This version should be used when there is good reason to believe variances are equal, such as when Levene's test is non-significant.

#### Welch's t-Test

Welch's t-test does not assume equal variances. It uses SE = sqrt(s1^2/n1 + s2^2/n2), and the test statistic is t = (x-bar1 - x-bar2) / SE, with degrees of freedom calculated using the Welch-Satterthwaite equation (usually resulting in a non-integer). Welch's test is more robust and is recommended as the default two-sample t-test.

<image>A side-by-side comparison of two independent groups. Panel A: Two overlapping histograms (or density curves) for Group 1 and Group 2, with vertical dashed lines at each group mean. The difference between means is annotated. Panel B: A summary table showing n, mean, SD for each group, the test statistic, df, and p-value. Panel C: A number line showing the 95% CI for the difference in means, with zero marked to indicate significance.</image>

### III. Paired t-Test

The paired t-test compares means when observations are naturally paired, such as before versus after treatment on the same patients or matched pairs in a twin study. The procedure calculates differences d_i = x_after,i - x_before,i for each pair, then computes d-bar and s_d. The test statistic is t = d-bar / (s_d / sqrt(n)), with df = n - 1, which is equivalent to a one-sample t-test on the differences. The paired design has the advantage of eliminating between-subject variability, thereby increasing statistical power.

As an example, blood glucose was measured before and after a dietary intervention in 12 patients. With d-bar = -15 mg/dL and s_d = 20 mg/dL, the test statistic is t = -15 / (20 / sqrt(12)) = -15 / 5.77 = -2.60, with df = 11 and a p-value of approximately 0.025 (two-sided). We reject H0 and conclude that the intervention significantly reduced blood glucose.

### IV. Assumptions and Diagnostics

The **normality** assumption requires that the data (or differences for the paired test) are approximately normally distributed. This can be checked with a histogram, Q-Q plot, or Shapiro-Wilk test. The t-test is robust to moderate violations when n is large (n >= 30). **Independence** requires that observations are independent within and between groups, which is violated if patients are clustered (for example, multiple measurements per patient). **Equal variances** are assumed only for the pooled t-test and can be checked with Levene's test or by comparing standard deviations (a rule of thumb is s_max / s_min < 2). When in doubt, use Welch's t-test. **Outliers** can heavily influence the mean and t-statistic and should be investigated; sensitivity analyses or nonparametric alternatives may be warranted.

<image>A diagnostic panel for t-test assumptions. Panel A: A Q-Q plot of residuals showing points approximately on the diagonal line -- normality assumption met. Panel B: Side-by-side boxplots of two groups with similar spreads -- equal variance assumption met. Panel C: A Q-Q plot with systematic curvature -- normality assumption violated. Panel D: Side-by-side boxplots with one group much more variable than the other -- unequal variances, use Welch's test. Each panel is annotated with the interpretation.</image>

### V. Effect Size: Cohen's d

The p-value depends on sample size, but effect size does not, making it an essential complement to significance testing. **Cohen's d** for two independent groups is calculated as d = (x-bar1 - x-bar2) / s_p, while for paired data it is d = d-bar / s_d. Cohen's guidelines provide benchmarks: d = 0.2 is a small effect, d = 0.5 is a medium effect, and d = 0.8 is a large effect. Always report effect size alongside the p-value for clinical interpretability.

### VI. Reporting Results

Results from t-tests should be reported in a standard format. An example: "The mean SBP in the treatment group (128.5 +/- 12.3 mmHg) was significantly lower than in the control group (135.2 +/- 14.1 mmHg); t(58) = -2.12, p = 0.038, Cohen's d = 0.51." The report should include descriptive statistics for each group (mean, SD, n), the test statistic and degrees of freedom, the exact p-value (not just "< 0.05"), the confidence interval for the difference, and an effect size measure. CONSORT guidelines for clinical trials recommend reporting CIs in addition to p-values.

<image>A forest plot showing the results of several t-test comparisons from a clinical study. Each row represents a different outcome variable (SBP, DBP, heart rate, BMI). For each, the point estimate (difference in means) is shown as a square, and the 95% CI is shown as a horizontal line. A vertical line at zero represents "no difference." Outcomes whose CIs exclude zero are highlighted. Sample sizes and p-values are listed to the right of each row.</image>

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