Choosing the right hypothesis test helps you turn data into a fair statistical decision. The best test depends on the type of response variable, the number of groups, and whether the groups are independent or paired. A flowchart is useful because it starts with simple questions and narrows the choices step by step.
This prevents using a test that does not match the study design or the kind of data collected.
For numerical data, t-tests compare means for one or two groups, while ANOVA compares means across three or more groups. For categorical data, chi-square tests often examine counts in categories, while proportion tests compare success rates. Paired designs, such as before-and-after measurements on the same people, require tests that account for the link between observations.
The goal is always to match the test statistic and assumptions to the structure of the data.
Key Facts
- One-sample t-test: tests whether a population mean differs from a hypothesized value, using t = (x̄ - μ0) / (s / √n).
- Two-sample t-test: compares the means of two independent groups, such as treatment versus control.
- Paired t-test: compares the mean of paired differences, using d̄ as the average difference within matched pairs.
- One-way ANOVA: compares means across three or more independent groups, using F = variation between groups / variation within groups.
- Chi-square test of independence: tests whether two categorical variables are associated, using χ² = Σ((O - E)² / E).
- Proportion tests: compare categorical success rates, such as one-proportion or two-proportion z-tests, using z = (p̂ - p0) / SE.
Vocabulary
- Hypothesis test
- A statistical procedure that uses sample data to evaluate a claim about a population.
- Response variable
- The outcome variable being measured or classified in a study.
- Independent groups
- Groups are independent when the observations in one group are not naturally linked to observations in another group.
- Paired data
- Paired data occur when two measurements are linked, such as measurements from the same person before and after a treatment.
- Categorical data
- Categorical data place observations into groups or labels, such as yes or no, red or blue, or treatment type.
Common Mistakes to Avoid
- Using a t-test for three or more groups: this is wrong because multiple pairwise t-tests increase the chance of a false positive, so a one-way ANOVA is usually the correct first test.
- Treating paired data as independent: this is wrong because matched observations share information, and ignoring the pairing can hide real effects or distort uncertainty.
- Using a chi-square test on means: this is wrong because chi-square tests are for counts in categories, not numerical averages.
- Choosing a test before identifying the response variable type: this is wrong because numerical responses, categorical responses, and counts often require different test families.
Practice Questions
- 1 A teacher compares the mean exam score of one class to a known district average of 75. The class has n = 30 students, x̄ = 78, and s = 6. Which hypothesis test should be used, and what is the test statistic?
- 2 A researcher compares recovery times for patients in 4 different treatment groups. The response is measured in days, and the groups are independent. Which test should be used to compare the mean recovery times?
- 3 A survey records whether students prefer online or in-person classes and whether they are freshmen, sophomores, juniors, or seniors. Explain which test should be used and why the data type matters.