Goodness-of-fit test idea
Big idea: Goodness-of-fit checks whether observed category counts match a claimed distribution.
Example: Is a die fair?
Compare observed rolls to expected equal frequencies.
Same chi-squared framework: You still use chi-squared formula.
The difference is one categorical variable with expected proportions.
Expected counts from proportions
Worked example
A spinner has 4 colors with expected proportions 0.1, 0.2, 0.3, 0.4.
After 200 spins, find expected counts.
Step by step
- Multiply total by each proportion
- E1=200×0.1=20, E2=40, E3=60, E4=80
- Check total expected = 200
Final answer
Expected counts are 20, 40, 60, 80.
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Chi-squared and degrees of freedom
Degrees of freedom: For goodness-of-fit with k categories and no estimated parameters, df = k - 1.
Worked example
There are 6 categories.
What is df?
Solution
- Use df = k - 1
- df = 6 - 1 = 5
Final answer
Degrees of freedom = 5.
IB-style question — χ² goodness-of-fit for normality
A teacher tests whether 200 exam marks follow N(60, 12²), grouping them into 5 classes.
(a) State the degrees of freedom. (b) Explain how each expected frequency is found.
Step by step
- (a) For a goodness-of-fit test, df = (number of classes) − 1. (At AI SL μ and σ are given, so there is no extra subtraction.)
- (b) Expected count for a class = total × P(that class) using normalcdf, with the two end classes open (lower = −1E99, upper = 1E99).
Final answer
(a) df = 4. (b) Eᵢ = 200 × normalcdf over each class (end classes use ±1E99).
[Diagram: math-normal-curve] - Available in full study mode
Interpretation and exam communication
Weak conclusion
- Only writes reject/fail reject
- No context
- No mention of model fit
Strong conclusion
- States decision and context
- Explains fit to claimed distribution
- Uses significance level language
Exam Tips:
- State H0 and H1 in words.
- Use p-value or critical-value rule consistently.
- Final sentence must reference context data.