In the full lesson
- T-Interval Formula (formula)
- Worked Example (example)
- When to Use t vs z (tip)
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
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.
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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