How to do a t-test and a χ² test on the Casio fx-CG50
Two-sample t-test (pooled), χ² test for independence from a contingency table and χ² goodness of fit. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
AI SL: χ² test for independence and goodness of fit, and the two-sample t-test (pooled, from data). AI HL adds one-sample tests and more. You must state H₀ and H₁, give the p-value (or the test statistic and critical value) and conclude in context.
Practise it: SL AI statistical testing · SL AI two-sample t-test · SL AI χ² tests · HL AI χ² and error types.
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
t-tests and χ² tests on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- t-test: MENU 2, sample A in List 1 and sample B in List 2. F3 (TEST) F2 (t) F2 (2-SAMPLE).
- Data: List, μ1: ≠ μ2, List(1): List1, List(2): List2, Freq 1 and 1, Pooled: On. Execute: read t, p and df.
- χ² independence: F3 (TEST) F3 (CHI) F2 (2WAY). Observed: Mat A. Press F2 (►MAT) to type the table into Mat A (set its size, e.g. 2 × 3), EXIT, then Execute. Read χ², p and df; the expected values are stored in Mat B.
- Goodness of fit: observed in List 1, expected in List 2 (expected frequencies, not proportions), then F3 (TEST) F3 (CHI) F1 (GOF). df = number of classes − 1 (minus any estimated parameters). Execute.
- p = 4.2056E−03 means 0.0042056: read the E−03.
- To see the expected frequencies: MENU 1, F3 (MAT/VCT), open Mat B.
t-tests and χ² tests on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- t-test: data in L1 and L2. stat → TESTS → 4: 2-SampTTest. Inpt: Data, List1 L1, List2 L2, μ1: ≠μ2, Pooled: Yes. Calculate.
- χ² independence: 2nd x⁻¹ (MATRIX) → EDIT → 1: [A], size 2 × 3, type the table. Then stat → TESTS → C: χ²-Test, Observed [A], Expected [B]. Calculate.
- Goodness of fit: observed in L1, expected in L2. stat → TESTS → D: χ²GOF-Test, Observed L1, Expected L2, df 3. Calculate.
- The expected table is written to [B]: view it in the MATRIX menu.
t-tests and χ² tests on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- t-test: data in two named columns. menu → Statistics → Stat Tests → 2-Sample t Test → Data. Choose the lists, Alternate Hyp μ1 ≠ μ2, Pooled: Yes. OK.
- χ² independence: store the table as a matrix (e.g. obs := 18,22,20][32,18,10 on a Calculator page), then menu → Statistics → Stat Tests → χ² 2-way Test, Observed Matrix: obs.
- Goodness of fit: menu → Statistics → Stat Tests → χ² GOF with the observed and expected lists and df.
- stat.expmatrix holds the expected frequencies after a 2-way test.
Worked example
(a) Sample A: 23.1, 25.4, 22.8, 26.0, 24.7, 25.9, 23.5, 24.2. Sample B: 21.9, 23.0, 22.4, 24.1, 21.5, 22.8, 23.3, 22.0. Test at 5% whether the means differ (pooled). (b) Observed table 18, 22, 20], [32, 18, 10: test for independence at 5%. (c) Observed 28, 35, 22, 15 against a uniform distribution.
Answers (checked with Python)
(a) t = 3.41266, p = 0.00420561 < 0.05: reject H₀, the means differ. (b) χ² = 7.65333, df = 2, p = 0.0217821 < 0.05: reject H₀ (not independent). The first expected frequency is 25. (c) χ² = 8.72, df = 3, p = 0.0332549 < 0.05: the data do not fit a uniform distribution.
Common calculator mistakes
- Pooled: the IB's two-sample t-test uses the pooled variance (Pooled: On/Yes). Leaving it off gives a slightly different p.
- Goodness of fit with proportions or percentages in the expected list: expected values must be frequencies that add up to the total.
- Wrong degrees of freedom: (rows − 1)(columns − 1) for independence; classes − 1 − (number of estimated parameters) for goodness of fit.
- Expected frequencies below 5: combine classes first (the calculator will not warn you).
- Concluding "H₀ is true": say there is insufficient evidence to reject H₀, and answer in the context of the question.
Do it in Desmos
Not in the exam. Desmos is not an approved calculator for IB exams, so never rely on it for Paper 2. It is excellent for learning, checking a GDC answer and building graphs for your Internal Assessment (IA).
- Make lists A and B of the two samples.
- ittest(A, B) runs a two-sample t-test and shows t and p; choose the alternative hypothesis in its menu. Check whether it is pooled: if its p differs slightly from your GDC's pooled test, that is why.
- Desmos has no built-in χ² test: work out Σ(O − E)² ÷ E with lists (O and E) to check the statistic.
The interactive graph loads here when you scroll to it.
Or open desmos.com/calculator and type:
A=[23.1,25.4,22.8,26,24.7,25.9,23.5,24.2]B=[21.9,23,22.4,24.1,21.5,22.8,23.3,22]ittest(A,B)O=[28,35,22,15]E=[25,25,25,25]total(frac{(O-E)^{2}}{E})
More calculator skills
Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the OS versions shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. Worked-example answers were recomputed independently in Python.