Module 5 · Inference · Topic 9

Test Statistics & P-values

Test Statistic. A test statistic measures how far the sample result is from the null hypothesis value, in standardized units. Common test statistics are z, t, and χ².

P-value. The p-value is the probability of getting a test statistic as extreme as (or more extreme than) the observed value, assuming H₀ is true. Small p-values provide evidence against H₀.

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Key formulas

General Form

Test statistic = (sample statistic − null value) / standard error

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

Interpreting P-values

If p-value = 0.03 and α = 0.05: since 0.03 < 0.05, we reject H₀. There is sufficient evidence to support H₁. If p-value = 0.12, we fail to reject H₀.

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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