Testing the fit of a model Problem set A Answers

20.0, 130, 102, 47.6

1. (a) \(X \sim B(3, 0.4)\)
2. (b) 54, 108, 72, 16

8, 32, 72, 128

1. (a) \(H_0\): heights follow \(N(20, 3^2)\); \(H_1\): they do not
2. (b) \(H_0\): the number faulty follows \(B(5, 0.3)\); \(H_1\): it does not
3. (c) df = 5

1. (a) 4
2. (b) χ² = 2.59, p = 0.629

1. (a) \(H_0\): the number of correct answers follows \(B(3, 0.4)\); E = 54, 108, 72, 16
2. (b) χ² = 5.26
3. (c) 5.26 < 7.815: do not reject \(H_0\). Insufficient evidence that the number of correct answers does not follow \(B(3, 0.4)\)

1. (a) 23.0, 45.9, 62.2, 45.9, 23.0
2. (b) \(H_0\): the journey times follow \(N(34, 5^2)\); \(H_1\): they do not; df = 4
3. (c) χ² = 8.38, p = 0.0787
4. (d) 0.0787 > 0.05, so do not reject \(H_0\). Insufficient evidence, at the 5% level, that the journey times do not follow \(N(34, 5^2)\).

1. (a) \(H_0\): the die is fair; \(H_1\): it is not
2. (b) E = 25; df = 5
3. (c) χ² = 6.4, p = 0.269
4. (d) 0.269 > 0.02, so do not reject \(H_0\). Insufficient evidence, at the 2% level, that the die is not fair.

1. (a) \(H_0\): the coins are fair, so \(X \sim B(5, 0.5)\); \(H_1\): they are not fair
2. (b) 10, 50, 100, 100, 50, 10
3. (c) df = 5; χ² = 27.03, p = 5.63 × 10⁻⁵
4. (d) 5.63 × 10⁻⁵ < 0.05, so reject \(H_0\). Sufficient evidence, at the 5% level, that the coins are not fair.

1. (a) 100, 80, 60, 40, 20
2. (b) \(H_0\): the number of books borrowed follows the model; \(H_1\): it does not; df = 4
3. (c) χ² = 6.23, p = 0.183; 0.183 > 0.05, so do not reject \(H_0\). Insufficient evidence, at the 5% level, that the model does not fit.