The Normal Distribution (AQA Level 3 Mathematical Studies (Core Maths): Paper 2A: Statistical Techniques): Exam Questions

The income tax, £$X$, paid last year by workers in a company can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$

Draw a line from each box on the left to the box with the most appropriate percentage.

Three quarters of the workers paid less than £15 000 income tax last year.

Select the correct statement about the gradient of the normal bell-shaped curve at $X$ = 15  000

One day, the time spent on social media by the students in a school is normally distributed with mean 140 minutes and standard deviation 25 minutes.

A student at the school is chosen at random.

Calculate the probability that the time the student spent on social media

(i) was under 175 minutes.

[2]

(ii) was above 130 minutes.

[2]

(iii) did not exceed 50% of the mean time.

[2]

The probability that the student spent between $t$ minutes and 170 minutes on social media is 0.3

Work out the value of $t$.

Scafell Pike in the Lake District is the highest mountain in England.

The maximum daily temperature (°C) in winter, $W$, at Scafell Pike can be modelled by a normal distribution with mean 2.5 and standard deviation 1.4

State the percentage of days in winter you would expect Scafell Pike to reach a maximum temperature within two standard deviations of the mean.

Give your answer to the nearest integer.

Calculate $P\left(W<1.5\right)$

Calculate $P\left(W>5.3\right)$

Calculate $P\left(-0.5<W<5.5\right)$

The maximum daily temperature (°C) in summer, $S$, at Scafell Pike can be modelled by a normal distribution with mean 12.7 and standard deviation 2.1

$P\left(S<x\right)=0.04$

Work out the value of $x$.

The normal distribution is one of the most important probability distributions in statistics.

Draw a line from each box on the left to the correct value on the right.

The heights, $X$ metres, of yucca plants at a garden centre are normally distributed with mean 1.58 and standard deviation 0.31.

Write the distribution in notation form.

Calculate $P\left(X<2\right)$

Calculate $P\left(X<1.3\right)$

60% of the yucca plants have a height of more than $k$ metres.

Work out the value of $k$

Research was carried out at a maternity hospital to find out the number of days that pregnancies last from conception to birth.

The data for this hospital’s patients can be modelled by a normal distribution with mean 275 days and standard deviation 12 days.

A patient at the hospital is chosen at random.

Calculate the probability that the number of days that the patient’s pregnancy lasts is

(i) less than 275

[1]

(ii) more than 290

[2]

(iii) within two standard deviations of the mean.

Give your answer as a decimal to four decimal places

[3]

At least 70% of pregnancies at the hospital did not exceed $n$ days, where $n$ is an integer.

Work out the lowest possible value of $n$

A research project was carried out at another maternity hospital.

At this hospital the number of days that pregnancies last is modelled by the distribution N(275, 15²)

80 patients in the project were expected to have their pregnancies last at least 270 days.

How many patients took part in the research project?

The annual salary of electrical technicians in the UK can be modelled by a normal distribution with mean £31 000 and standard deviation £7000 .

Based on this model, what is the median annual salary of an electrical technician in the UK?

An electrical technician is chosen at random.

Calculate the probability that the annual salary of this technician is

(i)more than £39 000

[2]

(ii) less than £26 000

[2]

(iii) between £26 000 and £39 000

[2]

Work out an estimate of the minimum annual salary of the 10% most highly paid electrical technicians in the UK.

Give your answer to the nearest £1000

A person’s total cholesterol level is the amount of cholesterol in their blood.

The UK government recommends that healthy adults should have a total cholesterol level below 5.0 millimoles per litre (mmol/l).

The total cholesterol level of an adult in the UK can be modelled by a normal distribution with mean 5.6 mmol/l and standard deviation 1.3 mmol/l.

Work out the probability that a randomly chosen adult in the UK has a total cholesterol level below 5.0 mmol/l.

Show that approximately one in every three adults in the UK has a total cholesterol level within 10% of the mean total cholesterol level.

Estimate the total cholesterol level exceeded by 75% of adults in the UK.