Module 5 · Inference · Topic 7

Hypothesis Testing Intro

Hypothesis Test. A hypothesis test is a formal procedure for using sample data to evaluate a claim about a population parameter. It assesses whether observed data is consistent with a stated hypothesis.

Steps of a Hypothesis Test. 1. State H₀ and H₁. 2. Choose a significance level α (commonly 0.05). 3. Collect data and compute the test statistic. 4. Find the p-value. 5. Make a decision: reject H₀ if p-value < α.

Significance Level. The significance level α is the probability of rejecting H₀ when it is actually true (Type I error rate). Common choices are 0.01, 0.05, and 0.10.

Type II Error and Power. A Type II error happens when we fail to reject H₀ even though H₁ is true. Its probability is β, and the power of a test is 1 − β: the probability of correctly rejecting a false H₀.

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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.

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When to use it

Decision Language

We either “reject H₀” or “fail to reject H₀.” We never say “accept H₀” because failing to find evidence against H₀ is not the same as proving it true.

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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