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
Nadia practises solving a puzzle cube. The time, $T$ seconds, that she takes to solve the cube on a randomly chosen attempt is modelled by a normal distribution with mean 48 and standard deviation 5
(a) Using standardisation, show that the probability that Nadia solves the cube in less than 45 seconds on a randomly chosen attempt is 0.2743

(2 marks)

On 20% of her attempts, Nadia takes longer than $t$ seconds.
(b) Use standardisation to find the value of $t$

(3 marks)

Nadia enters a competition. She may make up to 3 attempts, and she stops as soon as she solves the cube in less than 45 seconds. Her times on different attempts are independent.

Given that Nadia records a time of less than 45 seconds in the competition,
(c) find the probability that this time was recorded on her second attempt and was less than 40 seconds.

(5 marks)
Level 4 · 10 marksTry it
2
A farm sorts the potatoes it grows into five sizes by mass. The farm models the mass, $M$ grams, of one of its potatoes by $M \sim \mathrm{N}(180, 20^2)$

The sizes are defined in the table below, where $k$ is a constant.
Size Mini Small Medium Large Baker
Mass ( $m$ grams) $m < 140$ $140 \le m < 172$ $172 \le m < k$ $k \le m < 210$ $m \ge 210$
(a) Find the proportion of the farm's potatoes that are Small.

(3 marks)

The farm finds that 30% of its potatoes are Medium.
(b) Find the value of $k$ . Give your answer correct to 1 decimal place.

(4 marks)

A restaurant buys only Baker potatoes from the farm, and takes the heaviest 40% of them.
(c) Find the mass of the lightest potato that the restaurant buys, to 1 decimal place.

(5 marks)
Level 4 · 12 marksTry it
3
A factory fills bags with crisps. A quality controller takes a random sample of 160 bags from one day's production and records the mass, in grams, of crisps in each bag.

The masses are summarised in the box plot below.(a) Use linear interpolation to estimate the proportion of the bags in the sample that contained less than 26.6 grams of crisps.

(2 marks)

Using the quartiles of this sample, the factory decides to check the filling machine whenever a bag is found to contain at least $Q_3 + 1.5 \times (Q_3 - Q_1)$ grams of crisps.
(b) Find the smallest mass of crisps in a bag that would lead to the machine being checked.

(1 mark)

The quality controller proposes a normal distribution as a model for the mass of crisps in a bag filled at the factory.
(c) State, with a reason, whether the box plot supports the quality controller's proposal.

(1 mark)

The factory describes a bag as overfilled if the mass of crisps in it is more than 2.25 standard deviations above the mean mass.

Assuming that the mass of crisps in a bag is normally distributed,
(d) find the probability that a randomly chosen bag is described as overfilled.

(2 marks)

The mean mass of crisps in the bags filled at the factory is 25.9 grams. One bag contained 29.05 grams of crisps and was described as overfilled.
(e) Find the greatest value that the standard deviation of the mass of crisps in a bag could have.

(2 marks)
Level 4 · 8 marksTry it
4
A machine fills bottles with water. The volume of water in a bottle, $X$ ml, is modelled by $X \sim N(500, 8^2)$ .
(a) Find $P(X < 490)$ . (2 marks)
(b) Find the probability that a randomly chosen bottle contains within $10$ ml of the mean. (3 marks)
(c) A crate contains $5$ bottles, filled independently. Find the probability that every bottle in the crate contains at least $490$ ml. (3 marks)
(d) The machine is recalibrated: the mean stays at $500$ ml but the standard deviation, $\sigma$ ml, changes. Find the largest value of $\sigma$ for which $P(X < 490) \le 0.02$ . (4 marks)
Level 4 · 12 marksTry it
5
A machine fills jars with honey. The mass of honey in a jar is modelled by a normal distribution with mean $410$ g and standard deviation $12$ g.
(a) Find the probability that a randomly chosen jar contains more than $428$ g of honey. (2 marks)
(b) Determine the mass $m$ such that $99\%$ of jars contain less than $m$ grams, giving your answer to 3 significant figures. (3 marks)

A consumer group believes the machine has drifted and is now underfilling jars. The masses of a random sample of $40$ jars have mean $405.4$ g. The standard deviation is assumed to remain $12$ g.
(c) Test, at the $5\%$ significance level, whether there is evidence that the mean mass of honey per jar has fallen below $410$ g. State your hypotheses clearly, and state whether your conclusion would change at the $1\%$ level. (6 marks)
Level 4 · 11 marksTry it

Size	Mini	Small	Medium	Large	Baker
Mass ( $m$ grams)	$m < 140$	$140 \le m < 172$	$172 \le m < k$	$k \le m < 210$	$m \ge 210$

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