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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Every Year 11 student at Hillcrest School runs the same cross-country course during one week of the autumn term.

This year, the times taken by these students had a mean of $17.2$ minutes and a standard deviation of $2.2$ minutes, and $95\%$ of the times were between $12.9$ minutes and $21.5$ minutes.
(a) Comment on whether a normal distribution could be a suitable model for the time taken by a Year 11 student at Hillcrest School to complete the course this year.

(3 marks)
(b) Assume now that the time taken by a Year 11 student at Hillcrest School to complete the course can be modelled by a normal distribution with mean $17.2$ minutes and standard deviation $2.2$ minutes.

(i) Find the probability that a randomly chosen Year 11 student at Hillcrest School took $18$ minutes to complete the course.

(ii) Find the probability that a randomly chosen Year 11 student at Hillcrest School took between $15$ minutes and $20$ minutes to complete the course.

(iii) Two Year 11 students at Hillcrest School are selected at random.

Find the probability that each of them took between $15$ minutes and $20$ minutes to complete the course.

(3 marks)
(c) In the same week, a random sample of $30$ Year 11 students at Moorside School ran the same course. Their times, $t$ minutes, are summarised by

$\sum t = 549 \qquad \sum (t - \bar{t})^2 = 83.81$

Calculate estimates of the mean and the standard deviation of the times taken by Year 11 students at Moorside School.

(3 marks)
(d) Using your answers to part (c), compare the times taken by Year 11 students at the two schools.

(2 marks)
Level 5 · 11 marksTry it
2
A driving simulator records the time a driver takes to press the brake pedal after a hazard appears on the screen.

For drivers who are not distracted, this reaction time can be modelled by a normal distribution with mean $0.72$ seconds and standard deviation $0.08$ seconds.
(a) The diagram below shows the shape of this distribution.Label the values $0.72$ and $0.80$ on the horizontal axis.

(2 marks)
(b) State the probability that the reaction time of a driver who is not distracted is less than $0.72$ seconds.

(1 mark)
(c) Find the probability that the reaction time of a driver who is not distracted is more than $0.85$ seconds.

(1 mark)
(d) Find the probability that the reaction time of a driver who is not distracted is between $0.65$ seconds and $0.75$ seconds.

(1 mark)
(e) Determine the reaction time that is exceeded by $90\%$ of drivers who are not distracted.

(2 marks)
(f) A random sample of $20$ drivers took the simulator test while holding a conversation on a hands-free phone.

The sum of their $20$ reaction times was $15.12$ seconds.

Carry out a hypothesis test at the $5\%$ significance level to investigate whether the mean reaction time of drivers holding a hands-free phone conversation is greater than $0.72$ seconds.

You may assume that the reaction times of these drivers are normally distributed with standard deviation $0.08$ seconds.

(7 marks)
Level 5 · 14 marksTry it
3
The mass, $M$ kg, of fruit sold each day at a market stall is normally distributed with mean $84$ and standard deviation $9.5$ .
(a) The stall is open on $250$ days in a year. Find the expected number of these days on which the mass of fruit sold is more than $97$ kg.

(4 marks)

It is given instead that $M$ is normally distributed with mean $\mu$ and standard deviation $9.5$ , and that $\mathrm{P}(M > 97) = 0.18$ .
(b) Find the value of $\mu$ .

(3 marks)
(c) In this case, find the probability that more than $97$ kg of fruit is sold on fewer than $3$ of $10$ randomly chosen days.

(3 marks)
Level 5 · 10 marksTry it