- Asked 3/10
- 2069 Bhadra (old course) · 8 marks
Poker test, 1000 four-digit numbers (2069)
Short answer
Chi-square test statistic = 28.97. Critical value χ²0.10,4 = 7.78. Since 28.97 > 7.78, H0 (independence) is rejected at α=0.10; the numbers are not independent.
A sequence of 1000 four digit numbers has been generated and analysis indicates the following combinations with frequencies:
| Combination (i) | Observed Frequency (Oi) |
|---|---|
| Four different digits | 570 |
| One pair | 380 |
| Two pairs | 34 |
| Three like digits | 15 |
| Four like digits | 1 |
| 1000 |
Based on poker test check whether the numbers are independent. Use α=0.10 and N=4 is 7.78
Setup. H0: the numbers are independent. H1: they are not. N = 1000, five classes, df = 4, α = 0.10, χ²0.10,4 = 7.78.
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 | 570 | 504 | 8.643 |
| One pair | 0.432 | 380 | 432 | 6.259 |
| Two pairs | 0.027 | 34 | 27 | 1.815 |
| Three of a kind | 0.036 | 15 | 36 | 12.250 |
| Four of a kind | 0.001 | 1 | 1 | 0.000 |
χ0² = 28.97
Conclusion. Since 28.97 > 7.78, H0 is rejected at α = 0.10; the numbers are not independent. The largest contributions are "three like digits" (15 observed against 36 expected, 12.25) and "four different digits" (570 against 504, 8.64).
✓ Answer checked by Claude Opus 5.5 (high effort)
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