Chi-squared goodness of fit Problem set A Answers

\(H_0\): sales are in the ratio 5 : 3 : 2 (proportions 0.5, 0.3, 0.2). \(H_1\): they are not.

1. (a) 6
2. (b) 4
3. (c) 8

90, 60, 30, 20

27

1. (a) df = 4; E = 20
2. (b) χ² = 5.5, p = 0.24

1. (a) \(H_0\): the plays are in the stated proportions; \(H_1\): they are not
2. (b) df = 3; E = 160, 120, 80, 40
3. (c) χ² = 5.18, p = 0.159
4. (d) Do not reject \(H_0\). Insufficient evidence, at the 5% level, that the plays differ from the stated proportions.

1. (a) \(H_0\): equally popular; \(H_1\): not equally popular; E = 50
2. (b) χ² = 10.8
3. (c) 10.8 > 7.815: reject \(H_0\). Sufficient evidence that the candidates are not equally popular

1. (a) 20
2. (b) Yes: every count is within 4 of 20

1. (a) χ² = 69.5, p = 1.3 × 10⁻¹³, df = 5
2. (b) Reject \(H_0\): sufficient evidence that customers are not equally likely on each day
3. (c) Friday and Saturday are busier, for example more people shop at the end of the week

1. (a) χ² = 0.2, p = 0.999
2. (b) They are very close to the expected frequencies
3. (c) The results fit a fair die almost perfectly, closer than real rolls usually are. The student may have made up some of the results