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AS A-level Statistics · Edexcel 8MA0 · free A-level maths worksheet

The normal distribution

Properties of the normal curve, finding probabilities and values with a calculator, the standard normal distribution, finding an unknown mean or standard deviation.

12 original questions, 77 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

The mass, X grams, of a chocolate bar made by a machine is modelled by X ~ N(50, 42)

  1. Find P(X > 56)
  2. Find P(46 < X < 53)
  3. Find the value of a such that P(X < a) = 0.85
4 marks
Show answer
a) 0.0668 b) 0.6147 c) a = 54.15
2

The random variable Z ~ N(0, 12)

  1. Find P(Z < 1.2)
  2. Find P(−0.5 < Z < 1.5)
  3. Find the value of z such that P(Z > z) = 0.05
  4. Find the value of a such that P(−a < Z < a) = 0.8
4 marks
Show answer
a) 0.8849 b) 0.6247 c) z = 1.6449 d) a = 1.2816
3

The heights, H cm, of the sunflowers in a field are normally distributed. The curve of the probability density function of H has points of inflection at H = 150 and H = 190

  1. Write down the mean and the standard deviation of H
  2. Write down, to the nearest whole number, the percentage of sunflowers with heights between 130 cm and 210 cm.
  3. Find P(H > 200)
  4. Find the interquartile range of the heights.
6 marks
Show answer
a) mean 170 cm, standard deviation 20 cm b) 95% c) 0.0668 d) 27.0 cm
4

A machine cuts wooden dowels. The length, L cm, of a dowel is normally distributed with mean 12 cm and standard deviation σ cm.

Given that 3% of the dowels are shorter than 11.82 cm,

  1. show that σ = 0.096 to 3 decimal places.
  2. Find the proportion of dowels with lengths between 11.9 cm and 12.15 cm.
5 marks
Show answer
a) (11.82 - 12)/σ = -1.8808, so σ = 0.0957 = 0.096 (3 d.p.) b) 0.7921
5

The times, T minutes, taken by runners to complete a 5 km park run are modelled by a normal distribution with mean μ and standard deviation σ

20% of the runners take less than 25 minutes and 10% of the runners take more than 38 minutes.

  1. Find the value of μ and the value of σ. Show your working clearly.
  2. Find the probability that a randomly chosen runner takes between 28 and 32 minutes.
7 marks
Show answer
a) μ = 30.2, σ = 6.12 b) 0.256
6

The masses, M grams, of apples from an orchard are normally distributed with mean 150 g and standard deviation 12 g. Apples with masses between 140 g and 165 g are sold to a supermarket.

  1. Find the proportion of apples that are sold to the supermarket.
  2. An apple is chosen at random from those sold to the supermarket. Find the probability that its mass is more than 155 g.
5 marks
Show answer
a) 0.6920 b) 0.336
7

The mass, F grams, of flour in a bag filled by a machine is modelled by F ~ N(1010, 82)

  • A bag with less than 1000 g cannot be sold and is emptied and refilled, at a loss of 30p.
  • A bag with between 1000 g and 1025 g is sold at a profit of 40p.
  • A bag with more than 1025 g is sold at a profit of 25p.
  1. Find the proportion of bags that cannot be sold.
  2. Find the expected profit for every 1000 bags filled. Give your answer to the nearest pound.
6 marks
Show answer
a) 0.1056 b) £321
8

Kemi's journey time to work, J minutes, is modelled by J ~ N(42, 62). Her journeys on different days are independent.

  1. Find the probability that a journey takes more than 50 minutes.
  2. Find the probability that, in 20 working days, more than 2 of her journeys take more than 50 minutes.
  3. Kemi wants to arrive on time on 95% of days. Find how many minutes before her start time she should leave, giving your answer to the nearest minute.
7 marks
Show answer
a) 0.0912 b) 0.273 c) 52 minutes (t = 51.87)
9

At a café, the temperature of a cup of tea when served, C °C, is modelled by C ~ N(72, 42). The waiting time, W minutes, for a cup of tea is modelled by W ~ N(4.5, 1.22).

A customer is happy if the tea is served at between 65 °C and 80 °C and the waiting time is less than 6 minutes.

  1. Find the probability that a cup of tea is served at between 65 °C and 80 °C.
  2. State a necessary assumption about C and W for the manager to calculate the probability that a customer is happy.
  3. Find the probability that a randomly chosen customer is happy.
  4. Two customers order tea separately. Find the probability that at least one of them is happy.
7 marks
Show answer
a) 0.9372 b) C and W are independent c) 0.838 d) 0.974
10

A teacher models the time, T minutes, that students take to finish a short online quiz by T ~ N(6, 32)

  1. Find P(T < 0)
  2. Find P(T < 1 | T > 0)
  3. Using your answer to part (a), explain why this normal distribution may not be a good model for T

The teacher refines the model by only including values of T > 0

  1. Find the median time to finish the quiz using the refined model.
8 marks
Show answer
a) 0.0228 b) 0.0256 c) the model gives a probability of about 2.3% of a negative time, which is impossible d) 6.09 minutes
11

A machine fills cups of soup. The volume, V ml, in a cup is normally distributed with mean μ ml and standard deviation 5 ml. Given that 2.5% of cups contain more than 340 ml,

  1. find the value of μ

A cup is rejected if it contains less than 320 ml or more than 338 ml.

  1. Find the probability that a randomly chosen cup is rejected.
  2. A random sample of 12 cups is taken. Find the probability that fewer than 2 of them are rejected.
  3. The machine is adjusted so that only 1% of cups contain more than 340 ml. The standard deviation is unchanged. Find the new value of the mean.
9 marks
Show answer
a) μ = 330.2 b) 0.0801 c) 0.751 d) 328.4 ml
12

A display board uses 4 bulbs. The lifetime, L hours, of a bulb is modelled by L ~ N(800, 602). The display stops working as soon as any one of its bulbs fails. The lifetimes of the bulbs are independent.

A display is fitted with 4 new bulbs.

  1. Find the probability that a bulb lasts more than 750 hours.
  2. Find the probability that the display is still working after 750 hours.
  3. Given that the display is still working after 750 hours, find the probability that it is still working after 800 hours.
  4. Find the time, to the nearest hour, by which there is a 10% chance that the display has stopped working.
9 marks
Show answer
a) 0.7977 b) 0.405 c) 0.154 d) 683 hours

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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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