Choosing and critiquing statistical modelsAQA A-Level Maths: Subtopic test

1. 1
   A teacher models the number $X$ of pupils in a class of $30$ who walk to school by the binomial distribution $X\sim B(30,0.4)$.

   (a)

   Which of the following is an assumption needed for this binomial model?

   [1 mark]

   • AEach pupil's journey time is Normally distributed
   • BThe probability of walking is different for each pupil
   • CWhether one pupil walks is independent of whether any other pupil walks
   • DThe number of pupils in the class is not fixed

   (b)

   Pupils in this class often decide in friendship groups, so whether a pupil walks affects whether their friends walk. Which condition for the binomial model is not satisfied?

   [1 mark]

   • AThe trials are independent
   • BThe number of trials is fixed
   • CThere are only two outcomes for each pupil
   • DThe probability of success is $0.4$ for every pupil

   (c)

   The pupils in the class live at very different distances from the school. Explain why $B(30,0.4)$ may not be a suitable model.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A researcher records the monthly household income $X$ pounds in a city. The sample mean is $2400$, the sample median is $1900$ and the sample standard deviation is $1800$. She models $X$ by the Normal distribution $N(2400,1800^2)$.

   (a)

   Which feature of the sample data most strongly suggests that the Normal model may not be appropriate?

   [1 mark]

   • AThe mean is positive
   • BThe standard deviation is smaller than the mean
   • CIncomes are measured in pounds
   • DThe mean is greater than the median, suggesting positive skew

   (b)

   Using the model $N(2400,1800^2)$, the proportion of households predicted to have a negative income is closest to

   [1 mark]

   • A$0.0\%$
   • B$9.1\%$
   • C$50.0\%$
   • D$90.9\%$

   (c)

   State, with a reason, whether the Normal model is appropriate for monthly household income.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A call centre claims that $70\%$ of calls are answered within $20$ seconds. A manager records the next $10$ calls received on a Monday morning and models the number $X$ answered within $20$ seconds by $X\sim B(10,0.7)$.

   (a)

   State the distribution of $X$ and find the probability that at least $8$ of the $10$ calls are answered within $20$ seconds.

   [3 marks]

   (b)

   The manager's sample is the next $10$ calls on a Monday morning, when the call centre is unusually busy. Give two reasons, referring to this context, why the binomial model may not be appropriate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A lake contains a large population of a species of fish. A biologist measures a random sample of $200$ fish and finds that the sample mean length is $31$ cm and the sample standard deviation is $4$ cm. She proposes the model $L\sim N(31,4^2)$ for the length $L$ cm of a fish.

   (a)

   Of the $200$ fish in the sample, $27$ are longer than $38$ cm.
   (i) Use the model to find the probability that a fish is longer than $38$ cm.
   (ii) Find the number of fish longer than $38$ cm that the model predicts in a sample of $200$.
   (iii) Compare the prediction with the observed number and comment on whether the model is suitable.

   [6 marks]

   (b)

   Separately, a small tank holds $50$ fish, of which $10$ are undersized. A fisherman removes $10$ fish at random, one after another, without replacement. He models the number of undersized fish removed by $B(10,0.2)$.
   (i) Using this model, find the probability that none of the $10$ fish is undersized.
   (ii) Find the probability that the first two fish removed are both not undersized, using the actual numbers of fish in the tank, and compare it with the value given by the binomial model.
   (iii) Explain why the binomial model is not exact here, and state a change to the situation that would make it a good model.

   [6 marks]

   Total for question 4: 12 marks

End of questions