Goodness of fit when parameters are estimated Problem set A Answers

1. (a) 2.30
2. (b) Po(2.30); P(X = 2) = 0.265

1. (a) df = 4
2. (b) df = 6
3. (c) df = 5
4. (d) The second student. The first has df = groups − 1 and the second has df = groups − 2.

1. (a) x̄ = 37.4 min
2. (b) s = 10.3 min

1. (a) 2.28, 13.6, 34.1, 34.1, 13.6, 2.28
2. (b) x < 44, 44 ≤ x < 52, 52 ≤ x < 60, x ≥ 60 (4 groups)

1. (a) x̄ = 2.27, Po(2.27)
2. (b) X = 0, X = 1, X = 2, X = 3, X = 4, X ≥ 5; expected: 10.3, 23.5, 26.6, 20.1, 11.4, 8.03
3. (c) χ² = 13.6, df = 4, p = 0.00870: reject H0

1. (a) H0: the number of visits per day follows Po(1.8). H1: it does not.
2. (b) X = 0, X = 1, X = 2, X = 3, X ≥ 4; df = 4
3. (c) p = 0.692: do not reject H0

1. (a) χ² = 10.3, df = 3, p = 0.0165: reject H0
2. (b) p̂ = 0.646
3. (c) χ² = 2.73, df = 3, p = 0.435: do not reject H0

1. (a) s = 9.22
2. (b) x < 50, 50 ≤ x < 60, 60 ≤ x < 70, x ≥ 70; df = 2
3. (c) χ² = 8.50, df = 2, p = 0.0143: reject H0

1. (a) x̄ = 160 cm, s = 15.6 cm
2. (b) x < 130 cm, 130 ≤ x < 140 cm, 140 ≤ x < 150 cm, 150 ≤ x < 160 cm, 160 ≤ x < 170 cm, 170 ≤ x < 180 cm, x ≥ 180 cm; df = 7 − 1 − 2 = 4
3. (c) χ² = 11.4, df = 4, p = 0.0228: reject H0

1. (a) χ² = 2.13, df = 3, p = 0.545: do not reject H0
2. (b) χ² = 10.6, df = 4, p = 0.0311: reject H0

χ² = 1.24, df = 1, p = 0.266: do not reject H0

χ² = 16.1, df = 6, p = 0.0134: reject H0

1. (a) 2.00
2. (b) 2.28
3. (c) The mean 2.00 and the variance 2.28 are close, so a Poisson model is plausible.

1. (a) χ² = 6.17, df = 4, p = 0.187: do not reject H0
2. (b) χ² = 30.8, df = 4, p = 0.00000331: reject H0
3. (c) χ² is multiplied by 5 (6.17 to 30.8) while df stays 4, so the same departure from fair becomes significant in a larger sample.

1. (a) The chi-squared statistic is only reliable if every expected frequency is greater than 5. Two are not, so combine neighbouring groups first.
2. (b) The question is leading: it suggests the food is disappointing, so it does not measure pupils' real opinion fairly. The result is not valid.