Choosing a distribution and approximationsEdexcel A-Level Maths: Subtopic test

1. 1
   Each seed in a very large batch germinates independently with probability 0.45. A random sample of 60 seeds is planted and $X$ is the number that germinate.

   (a)

   Which distribution models $X$ exactly?

   [1 mark]

   • A$B(0.45,60)$
   • B$N(60,0.45)$
   • C$B(60,0.45)$
   • D$B(60,0.55)$

   (b)

   Which Normal distribution is the best approximation to the distribution of $X$?

   [1 mark]

   • A$N(27,12.15)$
   • B$N(27,27)$
   • C$N(27,3.85)$
   • D$N(27,14.85)$

   (c)

   Use a Normal approximation to estimate $P(X\ge30)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A school has many students, of whom 10% walk to school. A researcher selects the 25 students in one tutor group and records $X$, the number who walk to school. Students in the same tutor group often live in the same roads.

   (a)

   Which assumption of the binomial model for $X$ is most likely to fail?

   [1 mark]

   • AWhether one student walks is independent of whether any other student walks
   • BThe number of students observed is fixed at 25
   • CEach student either walks or does not walk
   • DThe sample is taken from a population of many students

   (b)

   Why would a Normal approximation to $X$ be unsuitable if a binomial model were used?

   [1 mark]

   • AEvery discrete variable is impossible to approximate by a Normal distribution
   • B$p=0.1$ is far from 0.5 and the mean $np=2.5$ is small, so the distribution is skewed
   • CThe mean 2.5 is not a whole number
   • DThe variance of a Normal distribution must exceed its mean

   (c)

   Suggest how the researcher could choose the sample so that a binomial model would be more suitable.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A biased coin has probability 0.55 of landing heads. It is tossed 80 times and $H$ is the number of heads.

   (a)

   Explain why $H$ can be modelled by a Normal distribution, and state the distribution that should be used.

   [3 marks]

   (b)

   Use your approximation to find $P(40\le H\le48)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A quality inspector takes samples of 50 items from production lines that make very large numbers of items, and counts the faulty items in each sample. Items are faulty independently of one another. On Line A the probability that an item is faulty is 0.48. On Line B it is 0.05.

   (a)

   Let $Y$ be the number of faulty items in a sample of 50 from Line A.
   (i) State two assumptions that are needed for $Y$ to follow a binomial distribution.
   (ii) State the Normal distribution that can be used to approximate $Y$.
   (iii) Use this approximation to find $P(Y\le20)$.

   [6 marks]

   (b)

   Let $X$ be the number of faulty items in a sample of 50 from Line B. A colleague suggests approximating $X$ by a Normal distribution to find $P(X\le1)$. Evaluate this suggestion.

   [6 marks]

   Total for question 4: 12 marks

End of questions