A-level · Statistical distributions

Finding normal probabilitiesExam-style questions

Finding P(X < a), P(X > a) and P(a < X < b) for a normal distribution as areas under the curve, and the inverse problem — finding the value x for a given probability (the inverse normal).

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35 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
The members of a swimming squad are timed over 50 m and over 100 m.

The time, $X$ seconds, for a member of the squad to swim 50 m is normally distributed with mean 32 and standard deviation 1.6
(a) (i) Using standardisation, find the probability that a randomly chosen member takes more than 34.4 seconds to swim 50 m.

(ii) Hence find the probability that a randomly chosen member takes between 29.6 and 34.4 seconds to swim 50 m.

(4 marks)

Six members of the squad are chosen at random.
(b) Find the probability that exactly 4 of these 6 members swim 50 m in less than 34.4 seconds.

(3 marks)

The time, $Y$ seconds, for a member of the squad to swim 100 m is normally distributed with mean 73 and standard deviation $\sigma$ . The 100 m time that is exceeded by 20% of the members is 76.37 seconds, to 2 decimal places.
(c) Show that $\sigma = 4.0$ to 2 significant figures.

(2 marks)

The constants $k$ and $p$ are such that

$\mathrm{P}(X > k) = p \qquad \text{and} \qquad \mathrm{P}(Y < 2k) = p$
(d) Find

(i) the value of $k$

(ii) the value of $p$

(5 marks)
Level 5 · 14 marksTry it
2
The distance, $D$ metres, that a golfer, Amir, hits the ball with his driver can be modelled by $D \sim \mathrm{N}(262, 10^2)$
(a) (i) Using standardisation, show that the probability that a randomly chosen drive by Amir travels more than 280 metres is 0.0359

(ii) The distance of each drive is recorded to the nearest metre. Find, using standardisation, the probability that a randomly chosen drive by Amir is recorded as 272 metres. You must show your working.

(7 marks)

The distance, $B$ metres, that a second golfer, Ben, hits the ball with his driver can be modelled by $B \sim \mathrm{N}(\mu, \sigma^2)$

Of Ben's drives, 2.28% travel more than 290 metres and 15.87% travel less than 251 metres.
(b) (i) Use this information to form two equations in $\mu$ and $\sigma$

(ii) Solve your equations to find the value of $\mu$ and the value of $\sigma$ . You must show your working.

(6 marks)
Level 5 · 13 marksTry it
3
The time, $T$ seconds, that an office printer takes to print a 20-page document is modelled by $T \sim \mathrm{N}(40, 2^2)$
(a) Find the proportion of 20-page print jobs that take less than 37.4 seconds.

(3 marks)
(b) (i) Find the lower quartile, $Q_1$ , of $T$

(ii) Write down the upper quartile, $Q_3$ , of $T$

(4 marks)
(c) Find the value of $Q_1 - 1.5 \times (Q_3 - Q_1)$ and the value of $Q_3 + 1.5 \times (Q_3 - Q_1)$

(3 marks)

An engineer treats a print time as an outlier if it lies outside the two values found in part (c). One 20-page print job is chosen at random.
(d) Show, using standardisation, that the probability of this print time being an outlier is 0.007, correct to 3 decimal places.

(3 marks)

The engineer finds that the time for this print job is not an outlier.
(e) Find the probability that this print job took more than 43 seconds. You must show your working.

(4 marks)
Level 5 · 17 marksTry it
4
Heights are modelled by $X \sim N(170, 64)$ cm.
(a) Find $P(X > 180)$ .
(b) Find $P(160 < X < 175)$ .
(c) Find the height exceeded by $10\%$ of people. Give probabilities to 4 d.p. and the height to 3 s.f.
Level 5 · 7 marksTry it
5
A confectioner makes bars of chocolate using a moulding machine.

The mass of a bar, $M$ $M$ grams, is normally distributed with
- a mean of $51.2$ g
- a standard deviation of $x$ g
The confectioner finds that $2\%$ $2%$ of the bars have a mass of less than $49.5$ $49.5$ g.
(a) Show that $x = 0.83$ $x = 0.83$ to 2 decimal places.

(2 marks)
(b) Find the proportion of the bars that have a mass between $50$ $50$ g and $52.5$ $52.5$ g.

(1 mark)

It costs the confectioner $21$ $21$ p to make each bar.
- Bars with $M < 50$ are sold to a discount shop for $30$ p each.
- Bars with $50 \leqslant M \leqslant 52.5$ are sold for $60$ p each.
- Bars with $M > 52.5$ are melted down and moulded again, at an extra cost of $7$ p per bar, and are then sold for $60$ p each.
(c) Find the expected profit made on $500$ $500$ of these bars. Give your answer to the nearest pound.

(5 marks)

Chocolate eggs are also made at the factory, in batches of several thousand.

Each egg, independently, has a probability of $0.025$ $0.025$ of being cracked.

From each batch, $100$ $100$ eggs are chosen at random and checked. The batch is accepted only if fewer than $5$ $5$ of these eggs are cracked.

The confectioner wants at least $90\%$ $90%$ of batches to be accepted.
(d) Explain whether this aim is likely to be met.

(4 marks)
Level 5 · 12 marksTry it