The two-sample t-test idea
Big idea: A two-sample t-test checks whether the MEANS of two groups are genuinely different, or whether the gap could just be chance.
Use it to compare two sets of measurements — for example the recovery times of runners on two training plans, or the heart rates of active vs non-active people.
t-test vs chi-squared: Use a t-test for the MEANS of numerical data.
Use chi-squared for COUNTS in categories.
Different data type → different test.
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Hypotheses and tails
Choosing the tail: ‘lower than’ / ‘greater than’ → one-tailed (H1: mu1 < mu2 or mu1 > mu2). ‘different from’ → two-tailed (H1: mu1 != mu2).
Worked example
A coach claims runners on Plan A have a LOWER mean recovery time than Plan B.
State H0 and H1.
Step by step
- H0: the means are equal -> muA = muB
- ‘lower’ signals a one-tailed test
- H1: muA < muB
Final answer
H0: muA = muB; H1: muA < muB (one-tailed).
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Running the test on the GDC
Let the GDC do the arithmetic: Enter both data lists, run 2-SampTTest, and read the p-value.
Assume the data are normally distributed.
Set Pooled: Yes unless the question says the variances differ.
Worked example
Two lists are entered. 2-SampTTest with H1: mu1 < mu2 gives p = 0.018.
Test at the 5% significance level.
Solution
- Compare p with the significance level alpha = 0.05
- 0.018 < 0.05
Final answer
p < alpha, so reject H0.
Decision rule: p < significance level -> reject H0 (the means differ as claimed). p >= level -> fail to reject H0.
Interpretation and exam communication
Weak conclusion
- Only writes ‘reject H0’
- No context
- Never restates the claim
Strong conclusion
- States the decision in context
- Says which mean is larger/smaller
- Uses the significance level
Exam tips
- State H0 and H1 in words AND symbols.
- Match the tail to the wording (‘lower/greater’ = one-tailed, ‘different’ = two-tailed).
- Never write ‘accept H1’ — write ‘reject H0’ or ‘fail to reject H0’.
- The final sentence must answer the original claim in context.