Module 5 · Inference · Topic 11

T-Test

Types of T-Tests. One-sample: compare a sample mean to a known value. Two-sample: compare two independent sample means. Paired: compare means of matched or before/after data.

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Key formulas

One-Sample T-Test Statistic

t = (x̄ − μ₀) / (s / √n), with df = n − 1

From the Inference formula sheet

Confidence Interval for a Mean: xbar +/- critical value * standard error — Estimate plus or minus a margin of error.
One-Sample Z Test: z = (xbar - mu_0) / (sigma / sqrt(n)) — Tests a sample mean against a null mean when population sigma is known.
Power: Power = 1 - beta — Probability of correctly rejecting a false null hypothesis.

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Worked example

Worked Example

H₀: μ = 100, H₁: μ ≠ 100. n = 25, x̄ = 104, s = 8. t = (104 − 100) / (8/5) = 4/1.6 = 2.50. With df = 24 and a two-tailed test, p-value ≈ 0.0196. Since 0.0196 < 0.05, reject H₀.

When to use it

Checking Assumptions

The t-test assumes the data comes from a normal population (or n is large) and is a random sample. Check for outliers and skewness, especially for small samples.

Related glossary terms

P-value: The probability of results this extreme or more extreme, assuming the null hypothesis is true. Example: A p-value of 0.03 is evidence against H0 at alpha = 0.05.
Confidence Level: The long-run success rate of a confidence interval procedure. Example: A 95% method captures the true parameter in about 95% of repeated samples.
Type I Error: Rejecting the null hypothesis when it is actually true. Example: The significance level alpha is the Type I error rate.
Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Example: A false negative in a hypothesis test is a Type II error.
Power: The probability of rejecting the null hypothesis when it is false. Example: Power equals 1 - beta, where beta is the Type II error probability.

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