A-level · Statistical distributions

The normal distributionExam-style questions

The normal model N(μ, σ²) for continuous data — its symmetric bell shape, mean = median = mode = μ, spread set by σ, and standardising any normal variable to the standard normal Z = (X − μ)/σ.

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16 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

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1
The times taken by a large group of athletes to run $400$ m are modelled by a normal distribution with mean $52.4$ seconds and standard deviation $2.8$ seconds.
(a) Find the probability that a randomly chosen athlete takes less than $50$ seconds, giving your answer to $3$ significant figures. (2 marks)
(b) Find the probability that a randomly chosen athlete takes between $50$ and $55$ seconds, giving your answer to $3$ significant figures. (3 marks)
(c) The fastest $15\%$ of athletes qualify for a final. Find the qualifying time, giving your answer to $1$ decimal place. (3 marks)
(d) Three athletes are chosen at random. Find the probability that exactly one of them takes less than $50$ seconds, giving your answer to $3$ significant figures. (3 marks)
Level 4 · 11 marksTry it
2
The masses, in grams, of apples from an orchard are normally distributed with mean $165$ g and standard deviation $12$ g.
(a) Find the probability that a randomly chosen apple has a mass greater than $180$ g. (2 marks)
(b) Find the probability that a randomly chosen apple has a mass between $150$ g and $175$ g. (3 marks)
(c) The heaviest $10\%$ of apples are graded as "large". Find the least mass of a large apple. (3 marks)
Level 4 · 8 marksTry it
3
The time taken for a courier to deliver a parcel is modelled as $X \sim N(30, 4^2)$ minutes.
(a) Find $P(X > 35)$ . (2 marks)
(b) Find the time $d$ such that $10\%$ of deliveries take less than $d$ minutes. (3 marks)

A rival company's delivery times are modelled as $Y \sim N(\mu, \sigma^2)$ . It is known that $P(Y < 20) = 0.25$ and $P(Y > 32) = 0.10$ .
(c) Find the values of $\mu$ and $\sigma$ , giving your answers to 3 significant figures. (6 marks)
(d) Give one reason why a normal distribution might be a suitable model for delivery times, and one potential limitation. (2 marks)
Level 4 · 13 marksTry it
4
The weights of apples from an orchard, $W$ grams, are modelled by a normal distribution.

It is known that $10\%$ of the apples weigh less than $120$ g, and $5\%$ of the apples weigh more than $190$ g.
(a) Find the value of the mean $\mu$ and the value of the standard deviation $\sigma$ , giving each answer to $3$ significant figures. (6 marks)
(b) An apple is classed as large if it weighs more than $175$ g.

Find the probability that a randomly chosen apple is large, giving your answer to $3$ significant figures. (2 marks)
(c) Find the weight that is exceeded by $25\%$ of the apples, giving your answer to the nearest gram. (3 marks)
Level 4 · 11 marksTry it