🎲 Goodness-of-Fit Test
The goodness-of-fit test answers: "Do my data fit a specific distribution?"
📚 The Idea
We compare what weexpected to see (theory) with what weactually observed (data).
If the difference is too large — we reject the hypothesis that the distribution fits.
📐 Formula
\(\chi^2 = \sum \frac{(O - E)^2}{E}\)
O = Observed | E = Expected
🔢 Example: Is the die fair?
We rolled a die 60 times and obtained:
| Outcome | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Observed (O) | 8 | 12 | 7 | 15 | 9 | 9 |
| Expected (E) | 10 | 10 | 10 | 10 | 10 | 10 |
Calculation:
χ² = (8-10)²/10 + (12-10)²/10 + (7-10)²/10 + (15-10)²/10 + (9-10)²/10 + (9-10)²/10
χ² = 0.4 + 0.4 + 0.9 + 2.5 + 0.1 + 0.1 = 4.4
χ² = 0.4 + 0.4 + 0.9 + 2.5 + 0.1 + 0.1 = 4.4
Degrees of freedom: df = k - 1 = 6 - 1 = 5
Critical value: χ²₀.₀₅,₅ = 11.07
Conclusion: 4.4 < 11.07 → do not reject H₀ → no evidence the die is unfair
⚠️ Conditions for use
All expected frequencies must be at least 5. If not — merge categories.