Goodness of fit tests and contingency tablesEdexcel A-Level Further Maths: Mind map

Exam questions on Goodness of fit tests and contingency tables

1. A six-sided die is rolled 120 times. The frequencies of the faces $1,2,3,4,5,6$ are $14, 25, 17, 22, 24, 18$ respectively. A test is carried out, at the 5% significance level, of 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 survey of 120 students recorded their gender and usual mode of travel to school. Of the 60 boys, 20 walk, 25 take the bus and 15 come by car. Of the 60 girls, 30 walk, 20 take the bus and 10 come by car. A $\chi^2$ test is used to investigate whether mode of travel is associated with gender.
   The test statistic is $3.56$, to 3 significant figures. State the conclusion of the test at the 5% significance level, giving the critical value used.2 marks
3. A farmer packs eggs in boxes of 4. In a random sample of 100 boxes, the numbers of boxes containing $0,1,2,3,4$ cracked eggs were $41, 38, 16, 4, 1$ respectively. The farmer suggests that the number of cracked eggs in a box follows a binomial distribution $\mathrm{B}(4,p)$, where $p$ is estimated from the data.
   Show that the estimate of $p$ is $0.215$, and find the expected frequency of boxes containing exactly one cracked egg, to 2 decimal places.3 marks