Goodness of fit and contingency tables

Sample questions

Original practice questions written in the style of the exam.

1. Question 15 marks

   A six-sided die is rolled 120 times with these results:

   Score	1	2	3	4	5	6
   Frequency	15	22	18	26	17	22

   Test, at the 5% level of significance, whether the die is fair.

   Mark scheme

   • B1 $H_0$: the die is fair (scores uniformly distributed); $H_1$: the die is not fair (1)
   • B1 Expected frequency 20 for each score (1)
   • M1 $\chi^2 = \sum \dfrac{(O - E)^2}{E} = \dfrac{25 + 4 + 4 + 36 + 9 + 4}{20}$ (1)
   • A1 $\chi^2 = 4.1$ (1)
   • A1 $\nu = 5$, critical value $11.070$; $4.1 < 11.070$ so do not reject $H_0$ — no evidence the die is unfair (1)
2. Question 26 marks

   150 students were asked which of three clubs, A, B or C, they preferred. The results, split by the stream each student studies, were:

   	A	B	C
   Science	30	20	10
   Arts	20	30	40

   Test, at the 5% level of significance, whether club preference is independent of stream.

   Mark scheme

   • B1 $H_0$: club preference and stream are independent; $H_1$: they are not independent (1)