- 2072 Magh · 8 marks
Poker test, 1000 four-digit numbers (2072 Magh)
Short answer
Using standard four‑digit probabilities, χ0² = 20.01. With 4 degrees of freedom and critical value 9.488 at α=0.05, since 20.01 > 9.488, the numbers are not independent.
A sequence of 1000 four digit numbers has been generated and an analysis indicates the following conditions. Based on poker test, are these numbers independent? (use α=0.05 and N = 4 is 9.488)
| Combination Distributions | Observed Frequency |
|---|---|
| 4 different Digits | 510 |
| 3 Like Digits | 40 |
| 4 Like Digits | 5 |
| 1 Pair | 410 |
| 2 Pair | 35 |
Setup. H0: the numbers are independent. H1: they are not. N = 1000, five patterns, df = 4, χ²0.05,4 = 9.488. The classes are rearranged in the usual order: 4 different, 1 pair, 2 pair, 3 like, 4 like.
Probabilities for four-digit numbers: 0.504, 0.432, 0.027, 0.036, 0.001.
| Pattern | Probability | Oi | Ei | (Oi − Ei)²/Ei |
|---|---|---|---|---|
| 4 different digits | 0.504 | 510 | 504 | 0.071 |
| 1 pair | 0.432 | 410 | 432 | 1.120 |
| 2 pair | 0.027 | 35 | 27 | 2.370 |
| 3 like digits | 0.036 | 40 | 36 | 0.444 |
| 4 like digits | 0.001 | 5 | 1 | 16.000 |
χ0² = 20.01
Conclusion. Since 20.01 > 9.488, H0 is rejected; the numbers are not independent according to the poker test. Most of the value comes from "4 like digits" (5 observed against 1 expected, contribution 16.0).
Remark: if the small classes (3 like and 4 like, Ei = 37) were merged, df = 3 and the statistic would be 5.29 < 7.815, so independence would be accepted. With the critical value 9.488 given, the answer is to reject.
✓ Answer checked by Claude Opus 5.5 (high effort)
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