Chi-squared goodness of fit testsEdexcel International A Level Further Maths: Mind map

Exam questions on Chi-squared goodness of fit tests

1. A die is thrown 120 times. Faces 1 to 6 occurred 14, 25, 17, 22, 24 and 18 times respectively. A $\chi^2$ goodness of fit test is to be used to test whether the die is fair.
   Calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$.2 marks
2. A firm records the number of machine breakdowns in each of 100 weeks. No breakdowns occurred in 27 weeks, 1 in 35 weeks, 2 in 22 weeks, 3 in 11 weeks, 4 in 4 weeks and 5 in 1 week. The mean number of breakdowns per week is 1.33. The firm models the weekly number of breakdowns by a Poisson distribution with mean 1.33. The expected frequencies are 26.45, 35.18, 23.39, 10.37, 3.45 and 1.17 for 0, 1, 2, 3, 4 and 5 or more breakdowns respectively.
   Calculate the value of the test statistic for the combined cells.2 marks
3. An inspector tests 120 boxes, each containing 4 bulbs, and records the number of defective bulbs in each box. There were 48 boxes with no defective bulbs, 42 with 1, 22 with 2, 7 with 3 and 1 with 4. The inspector suggests that the number of defective bulbs in a box can be modelled by $B(4,p)$, with $p$ estimated from the data.
   State suitable hypotheses and show that the estimate of $p$ is 0.231 to 3 significant figures.3 marks