Goodness of fit when parameters are estimated Problem set B Answers

1. (a) 3.34
2. (b) Po(3.34); P(X = 2) = 0.198

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

1. (a) x̄ = 165 cm
2. (b) s = 8.23 cm

1. (a) 7.17, 23.2, 44.7, 44.7, 23.2, 6.59, 0.575
2. (b) x < 30 cm, 30 ≤ x < 35 cm, 35 ≤ x < 40 cm, 40 ≤ x < 45 cm, 45 ≤ x < 50 cm, x ≥ 50 cm (6 groups)

1. (a) x̄ = 2.01, Po(2.01)
2. (b) X = 0, X = 1, X = 2, X = 3, X = 4, X ≥ 5; expected: 13.4, 26.9, 27.1, 18.1, 9.11, 5.36
3. (c) χ² = 2.88, df = 4, p = 0.579: do not reject H0

1. (a) H0: the number of orders per hour follows Po(3.4). H1: it does not.
2. (b) X ≤ 1, X = 2, X = 3, X = 4, X = 5, X ≥ 6; df = 5
3. (c) p = 0.0299: reject H0

1. (a) χ² = 17.2, df = 6, p = 0.00848: reject H0
2. (b) p̂ = 0.527
3. (c) χ² = 12.0, df = 4, p = 0.0171: reject H0

1. (a) s = 9.45 min
2. (b) x < 36 min, 36 ≤ x < 44 min, 44 ≤ x < 52 min, x ≥ 52 min; df = 2
3. (c) χ² = 2.22, df = 2, p = 0.330: do not reject H0

1. (a) x̄ = 36.9 min, s = 6.34 min
2. (b) x < 30 min, 30 ≤ x < 35 min, 35 ≤ x < 40 min, 40 ≤ x < 45 min, x ≥ 45 min; df = 5 − 1 − 2 = 2
3. (c) χ² = 1.24, df = 2, p = 0.538: do not reject H0

1. (a) χ² = 3.07, df = 3, p = 0.381: do not reject H0
2. (b) χ² = 5.60, df = 5, p = 0.347: do not reject H0

χ² = 1.32, df = 2, p = 0.517: do not reject H0

χ² = 7.33, df = 6, p = 0.292: do not reject H0

1. (a) 2.39
2. (b) 5.17
3. (c) The variance 5.17 is much larger than the mean 2.39 (about 2.16 times). A Poisson distribution has mean = variance, so a Poisson model is not plausible.

1. (a) χ² = 6.40, df = 5, p = 0.269: do not reject H0
2. (b) χ² = 32.0, df = 5, p = 0.00000594: reject H0
3. (c) χ² is multiplied by 5 (6.40 to 32.0) while df stays 5, 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 sample is not representative: gym members exercise more than most pupils, so the results do not measure the exercise of all pupils. The result is not valid.