Statistics 3 (S3): Goodness of fit and contingency tablesEdexcel International A Level Further Maths: Topic test

1. 1
   A lottery machine is meant to select the digits $0$ to $9$ with equal probability. Over $200$ draws, the digits $0$ to $9$ occurred $18$, $24$, $19$, $21$, $17$, $23$, $22$, $16$, $20$ and $20$ times respectively. A $\chi^2$ goodness of fit test is to be carried out at the $5\%$ significance level.

   (a)

   How many degrees of freedom should be used?

   [1 mark]

   • A$9$
   • B$10$
   • C$8$
   • D$199$

   (b)

   Which is the null hypothesis?

   [1 mark]

   • AThe observed frequencies are all equal to $20$
   • BThe observed and expected frequencies are different
   • CThe digits are drawn with equal probability, so a discrete uniform distribution is a suitable model
   • DEach digit depends on the previous digit drawn

   (c)

   Calculate the value of the $\chi^2$ test statistic.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A seed company claims that each seed germinates with probability $0.7$. In an experiment, $120$ trays each hold $5$ seeds, and the number of seeds that germinate in each tray is recorded. The numbers of trays with $0$, $1$, $2$, $3$, $4$ and $5$ seeds germinating were $1$, $7$, $20$, $38$, $35$ and $19$ respectively. A $\chi^2$ test is to be carried out of whether the number germinating in a tray follows $\mathrm{B}(5,\,0.7)$.

   (a)

   Find the expected number of trays in which exactly $4$ seeds germinate.

   [1 mark]

   • A$24.0$
   • B$43.2$
   • C$37.0$
   • D$20.2$

   (b)

   Suppose the probability $0.7$ had not been given but had been estimated from the data, and the cells are combined so that there are $4$ cells. How many degrees of freedom should be used?

   [1 mark]

   • A$3$
   • B$4$
   • C$1$
   • D$2$

   (c)

   The expected frequencies for $0$, $1$ and $2$ seeds germinating are $0.29$, $3.40$ and $15.88$ (to $2$ d.p.). Explain which cells must be combined and why.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of cars arriving at a toll booth in each of $120$ one-minute intervals was recorded. The numbers of intervals with $0$, $1$, $2$, $3$, $4$ and $5$ or more cars were $15$, $35$, $31$, $24$, $9$ and $6$ respectively. The total number of cars arriving in the $120$ intervals was $240$. A $\chi^2$ test is to be carried out of whether a Poisson distribution is a suitable model.

   (a)

   Find the mean of the Poisson distribution that should be used, and calculate the expected frequency for exactly $2$ cars.

   [3 marks]

   (b)

   The expected frequencies for $0$, $1$, $2$, $3$, $4$ and $5$ or more cars are $16.24$, $32.48$, $32.48$, $21.65$, $10.83$ and $6.32$ respectively. Carry out the test at the $5\%$ significance level, stating your hypotheses.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The reaction times of $200$ drivers to a warning light were recorded, in seconds. The numbers of drivers with times in the classes less than $0.40$, $0.40$ to less than $0.46$, $0.46$ to less than $0.52$, $0.52$ to less than $0.58$, $0.58$ to less than $0.64$ and $0.64$ or more were $14$, $31$, $53$, $49$, $36$ and $17$ respectively. The sample mean is $0.52$ and the sample standard deviation is $0.08$.

   (a)

   Test, at the $5\%$ significance level, whether the reaction times can be modelled by a Normal distribution with mean and standard deviation equal to the sample values.

   [6 marks]

   (b)

   The manufacturer of the warning light states that reaction times follow a Normal distribution with mean $0.50$ and standard deviation $0.10$. Test this claim at the $5\%$ significance level, and say how the degrees of freedom differ from those in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A courier company claims that its deliveries to a business park are made at times uniformly distributed between 9 am and 1 pm. Of $160$ deliveries, $52$ were made between 9 am and 10 am, $38$ between 10 am and 11 am, $30$ between 11 am and 12 noon, and $40$ between 12 noon and 1 pm. A $\chi^2$ test is to be carried out at the $5\%$ significance level.

   (a)

   What is the expected number of deliveries between 10 am and 11 am?

   [1 mark]

   • A$40$
   • B$53.3$
   • C$38$
   • D$160$

   (b)

   Find the value of the $\chi^2$ test statistic.

   [1 mark]

   • A$248$
   • B$6.2$
   • C$0.6$
   • D$0.155$

   (c)

   Using your value of the test statistic from part (b), state the number of degrees of freedom and the conclusion of the test, in context.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sixth-form college records how $180$ students travel to college. Of the $90$ Year 12 students, $30$ travel by bus, $22$ by bicycle and $38$ on foot. Of the $90$ Year 13 students, $40$ travel by bus, $12$ by bicycle and $38$ on foot. A $\chi^2$ test is to be carried out at the $5\%$ significance level to see whether the method of travel is associated with year group.

   (a)

   What is the expected number of Year 13 students who travel by bicycle, if there is no association?

   [1 mark]

   • A$12$
   • B$34$
   • C$30$
   • D$17$

   (b)

   How many degrees of freedom should be used?

   [1 mark]

   • A$5$
   • B$3$
   • C$2$
   • D$6$

   (c)

   Write down the null hypothesis, in context. Given that the test statistic is $4.37$, state the conclusion of the test.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The numbers of girls in $150$ families, each with exactly $4$ children, were recorded. The numbers of families with $0$, $1$, $2$, $3$ and $4$ girls were $9$, $41$, $56$, $36$ and $8$ respectively. A $\chi^2$ test is to be carried out of whether a binomial distribution $\mathrm{B}(4,\,p)$ is a suitable model, with $p$ estimated from the data.

   (a)

   Find the estimate of $p$ and calculate the expected number of families with exactly $2$ girls.

   [3 marks]

   (b)

   The expected frequencies for $0$, $1$, $3$ and $4$ girls are $10.28$, $39.25$, $35.75$ and $8.53$ respectively. Carry out the test at the $5\%$ significance level.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A teacher gives $100$ students a quiz of $6$ questions and records the number of questions each student answers correctly. The numbers of students scoring $0$, $1$, $2$, $3$, $4$, $5$ and $6$ were $3$, $10$, $21$, $30$, $24$, $10$ and $2$ respectively.

   (a)

   Test, at the $5\%$ significance level, whether the number of correct answers can be modelled by $\mathrm{B}(6,\,p)$, where $p$ is estimated from the data.

   [6 marks]

   (b)

   The teacher believes that each student has probability $0.4$ of answering each question correctly. Test, at the $5\%$ significance level, whether the scores follow $\mathrm{B}(6,\,0.4)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions