Why this chapter matters
Hypothesis tests provide a structured way to judge whether sample evidence is strong enough to challenge a baseline claim.
What you will learn
- Write valid null and alternative hypotheses for one-parameter questions.
- Compute and interpret p-values in one-sample z or t tests.
- Distinguish statistical significance from practical importance.
Understand the core ideas
Hypothesis testing provides a repeatable framework for deciding whether sample evidence conflicts with a baseline claim. You state a null hypothesis that fixes a parameter value, then an alternative that describes the direction or kind of difference of interest. A test statistic measures how far the observed sample result is from what the null would predict, after scaling by expected variability. The p-value is then computed under the assumption that the null is true. Smaller p-values indicate stronger incompatibility between observed data and the null model.
Decision language must stay precise. Rejecting the null at level alpha controls long run false positive rate in repeated testing, but it does not prove the alternative is true with certainty. Failing to reject the null is not proof of no effect; it may reflect limited sample size or noisy data. Practical significance is also separate from statistical significance, so effect size and context should be reported with the test outcome. This keeps conclusions useful and avoids overclaiming.
Key terms
- Null hypothesis
- Baseline claim about a population parameter used for probability calculations.
- Alternative hypothesis
- Competing claim indicating direction or presence of a difference.
- P-value
- Probability under null model of data at least as extreme as observed.
- Significance level
- Prechosen cutoff alpha that controls long run Type I error rate.
One-proportion z test
Test H0: p = 0.70 versus Ha: p is not 0.70, with n = 200 and x = 154 successes.
- Compute sample proportion p-hat = 154/200 = 0.77.
- Compute null standard error approximately 0.0324.
- Compute z = (0.77 - 0.70) / 0.0324 approximately 2.16.
- Find two-sided p-value about 0.031 and compare to alpha 0.05.
A common misconception
Claim: The p-value is the probability that the null hypothesis is true.
Correction: The p-value is computed assuming the null is true; it measures extremeness of data, not probability of truth of hypotheses.
Lessons in this chapter
- Hypothesis testing frameworkFollow a full test workflow from claim to decision.
- One-sample tests for meansRun and interpret one-sample t tests with assumptions.
- One-sample tests for proportionsRun and interpret one-proportion z tests.
- Reading p-values and errorsExplain Type I and Type II errors in context. Read the full guide →
Study task
Chapter checkpoint
A test returns p = 0.03 at alpha = 0.05. What is the decision?
Reject the null hypothesis because the p-value is below the significance level.