Goodness of fit tests and contingency tablesEdexcel A-Level Further Maths: Subtopic test

1. 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.

   (a)

   Find the expected frequency of each face if the die is fair.

   [1 mark]

   • A$6$
   • B$20$
   • C$24$
   • D$120$

   (b)

   State the number of degrees of freedom for the test.

   [1 mark]

   • A$6$
   • B$4$
   • C$5$
   • D$119$

   (c)

   Calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   Find the expected number of boys who take the bus, assuming that mode of travel is independent of gender.

   [1 mark]

   • A$22.5$
   • B$25$
   • C$20$
   • D$45$

   (b)

   State the number of degrees of freedom for this test.

   [1 mark]

   • A$6$
   • B$1$
   • C$3$
   • D$2$

   (c)

   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]

   Total for question 2: 4 marks

3. 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.

   (a)

   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]

   (b)

   The other expected frequencies are $37.97$ for 0, $17.09$ for 2, $3.12$ for 3 and $0.21$ for 4 cracked eggs. Test, at the 5% significance level, whether $\mathrm{B}(4,p)$ is a suitable model. State your hypotheses.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A weaver inspects 100 one-metre lengths of cloth and records the number of faults in each. The numbers of lengths with $0, 1, 2, 3$ and 4 or more faults were $30, 36, 20, 9$ and $5$ respectively. The 5 lengths with 4 or more faults contained 21 faults in total, so the sample mean number of faults per length is $1.24$. The weaver suggests that the number of faults per metre follows a Poisson distribution.

   (a)

   The expected frequencies for a Poisson distribution with mean $1.24$ are $28.94, 35.88, 22.25$ and $12.93$ for $0,1,2$ and 3 or more faults. Test, at the 5% significance level, whether a Poisson model with mean $1.24$ is suitable. State your hypotheses and the degrees of freedom.

   [6 marks]

   (b)

   The manufacturer claims instead that the number of faults per metre follows a Poisson distribution with mean $1.5$, with expected frequencies $22.31, 33.47, 25.10, 12.55$ and $6.56$ for $0,1,2,3$ and 4 or more faults. Test this claim at the 5% significance level, and explain why the degrees of freedom differ from part (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions