Module 5 · Inference · Topic 2

Confidence Intervals Intro

Confidence Level. The confidence level (e.g., 95%) is the long-run proportion of intervals that would capture the true parameter if we repeated the sampling process many times.

Margin of Error. The margin of error (ME) is the half-width of the interval. It depends on the critical value (from the z or t distribution), the standard deviation, and the sample size.

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Key formulas

Margin of Error (z-based)

ME = z* × (σ / √n)

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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When to use it

Common Critical Values

For a 90% CI: z* = 1.645. For 95%: z* = 1.96. For 99%: z* = 2.576. Higher confidence → larger z* → wider interval.

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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