- Asked 3/10
- 2070 Magh · 7 marks
Poker test, 1000 four-digit numbers (2070 Magh)
Short answer
The chi‑square statistic is 39.19, with 4 degrees of freedom and critical value 9.49 at α=0.05. Since 39.19 > 9.49, the hypothesis of independence is rejected.
A sequence of 1000 four digit numbers has been generated and analysis indicates the following combinations and frequencies:
| Combination (i) | Observed Frequency (Oi) |
|---|---|
| Four different digits | 540 |
| One pair | 380 |
| Two pairs | 54 |
| Three like digits | 25 |
| Four like digits | 1 |
| 1000 |
Based on poker test check whether the numbers are independent. Use α=0.05 and χ² = 9.49
Setup. H0: the numbers are independent. H1: they are not. N = 1000, five classes, df = 4, α = 0.05, χ²0.05,4 = 9.49.
Probabilities for four-digit numbers: 0.504, 0.432, 0.027, 0.036, 0.001. Ei = 1000 × probability.
| Pattern | Probability | Oi | Ei | (Oi − Ei)²/Ei |
|---|---|---|---|---|
| Four different | 0.504 | 540 | 504 | 2.571 |
| One pair | 0.432 | 380 | 432 | 6.259 |
| Two pairs | 0.027 | 54 | 27 | 27.000 |
| Three of a kind | 0.036 | 25 | 36 | 3.361 |
| Four of a kind | 0.001 | 1 | 1 | 0.000 |
χ0² = 39.19
Conclusion. Since 39.19 > 9.49, H0 is rejected; the numbers are not independent. The biggest contribution is from "two pairs" (54 against 27 expected, contribution 27.0).
✓ Answer checked by Claude Opus 5.5 (high effort)
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