Module 5 · Inference · Topic 4

CI for Means (σ unknown)

The t-Distribution. The t-distribution is similar to the standard normal but has heavier tails. It accounts for the extra uncertainty from estimating σ with s. As n grows, t approaches z.

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Key formulas

T-Interval Formula

x̄ ± t* × (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

n = 20, x̄ = 85, s = 6. For a 95% CI with df = 19: t* ≈ 2.093. ME = 2.093 × (6/√20) = 2.093 × 1.342 = 2.81. CI = (82.19, 87.81).

When to use it

When to Use t vs z

Use z when σ is known (rare in practice). Use t when σ is unknown and you estimate it with s (the common case). The t-interval is the workhorse for real data.

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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