t-test (AI HL)
The t-test checks a claim about a population mean when the population standard deviation isn't known - which, on the GDC paper, is almost always the case, since it's estimated from the sample data instead. This page covers the one-sample and paired versions, the decision rule that turns a p-value into a conclusion, and the mistakes that cost marks. It's part of the broader Hypothesis Testing topic.
18 questions on this sub-topic.
The decision rule
Covered under IB syllabus reference SL4.11: formulation of \(H_0\) and \(H_1\), significance levels and p-values. Your GDC runs the t-test itself; you supply the hypotheses and the conclusion.
Decision rule
Reject \(H_0\) if \(p <\) significance level
Not in the formula booklet - this is the rule that applies to every hypothesis test on the course, not just the t-test.
Degrees of freedom (one-sample)
\(\nu = n-1\)
Not in the formula booklet. The GDC works this out for you when you enter the sample size, but you should be able to state it.
Need the full syllabus wording and formula-booklet reference table? See Hypothesis Testing.
Worked examples
A two-tailed test gives a p-value of 0.08.
(a) State whether \(H_0\) is rejected at the 5% level.
(b) If the equivalent one-tailed test were appropriate and the sample is in the predicted direction, find the one-tailed p-value and state the conclusion at the 5% level.
Worked solution
(a) \(0.08 > 0.05\), so do not reject \(H_0\). M1 A1
(b) One-tailed \(p = \dfrac{0.08}{2}\) M1
\(= 0.04 < 0.05\), so reject \(H_0\). A1
A sample of \(n=20\) has mean \(\bar x=52\) and \(s=6.\) Test, at the 5% level, whether the population mean exceeds 50.
(a)(i) State \(H_0\) and \(H_1.\)
(a)(ii) Find the p-value.
(a)(iii) State the conclusion, in context.
Worked solution
(a)(i) \(H_0:\mu=50.\) A1
\(H_1:\mu>50.\) A1
(a)(ii) One-sample \(t\)-test on the GDC with \(\bar x=52,\ s=6,\ n=20\) against \(\mu=50\): M1
Step 2 - State the p-value.
\(p\approx0.076.\) A1
(a)(iii) Since \(p=0.076>0.05\), R1 do not reject \(H_0\): insufficient evidence the mean exceeds 50. A1
Common mistakes
- Forgetting to state the conclusion in context. "Reject \(H_0\)" alone rarely earns full marks - the final answer needs to say what that means for the actual situation, e.g. "there is evidence the mean exceeds 50."
- Using the wrong t-test for paired data. If the same individuals are measured twice (before/after), it's a paired t-test on the differences, not a two-sample test on two independent groups.
- Mixing up one-tailed and two-tailed p-values. A two-tailed p-value must be halved before comparing it to a one-tailed significance level, and only when the sample result is in the predicted direction.
Ready to practise properly?
16 t-test questions, marked instantly like the real exam.
Quick answers
When do I use a t-test instead of a z-test?
Use a t-test whenever the population standard deviation is unknown and is estimated from the sample - this is almost every t-test question you'll see on the GDC paper.
What is the decision rule for a hypothesis test?
Reject \(H_0\) if the p-value is less than the stated significance level; otherwise do not reject \(H_0\). See the parent topic's GDC guidance for how to run the test itself.