A LevelMathsAQAStatisticsExam QuestionsStatistical DistributionsWorking with Distributions

Working with Distributions (AQA A Level Maths: Statistics): Exam Questions

Exam code: 7357

2 hours20 questions
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Medium
Hard
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1
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3 marks

George throws a ball at a target 15 times.

Each time George throws the ball, the probability of the ball hitting the target is 0.48.

The random variable X represents the number of times George hits the target in 15 throws.

Find

(i) space straight P open parentheses X equals 3 close parentheses

(ii) space straight P open parentheses X greater or equal than 5 close parentheses

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2
4 marks

The table below shows five scenarios involving different random variables. Complete the table by placing a cross (×) in the correct box to show whether each random variable can be modelled by a binomial distribution, a normal distribution, or neither.

The first row has been completed for you.

Scenario

Binomial

Normal

Neither

The digits 1 to 9 are written on individual counters and placed in a bag. A child randomly selects one counter. The random variable A represents the number written on the counter.

×

A farmer has many hens. The random variable B represents the mass, in kg, of a randomly selected hen.

A fair coin is flipped 100 times. The random variable C represents the number of times it lands on tails.

A teacher has a 30-minute lunch break. The random variable D represents the number of emails he receives during his lunch break.

In a class of 30 students, each student rolls a fair six-sided die. The random variable E represents the number of students who roll a number less than 5.

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3a
2 marks

In an experiment there are a fixed number of trials and each trial results in either a success or a failure. Let X be the number of successful trials.

Write down the two further conditions required for X to follow a binomial distribution.

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3b
3 marks

A fair spinner has 8 sectors labelled 1 to 8. For each of the following, give a reason to explain why a binomial distribution would not be an appropriate model for the random variable.

(i) The random variable A is the number of times the spinner is spun until it first lands on 1.

(ii) When the spinner is spun it always rotates exactly 115°. The random variable B is the number of times the spinner lands on 1 in 20 spins.

(iii) The random variable C is the number on the sector the spinner lands on when it is spun once.

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4a
3 marks

For each of the following, state with a reason whether the random variable is discrete or continuous.

(i) 100 red squirrels are sampled. The random variable A is the tail length, in cm, of a randomly selected squirrel.

(ii) 100 students sit a test marked out of 50. The random variable B is the number of marks scored by a randomly selected student.

(iii) 100 men are in a shoe shop. The random variable C is the shoe size of a randomly selected man.

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4b
1 mark

The histogram below shows the distribution of measurements of a random variable D.

Histogram shows relative frequency density with a U-shape distribution. Bars rise on both sides with a dip in the centre; x-axis labelled 'Measurements of D'.

State, with a reason, whether a normal distribution would be an appropriate model for D.

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5a
1 mark

A fair spinner has 8 sectors labelled 1 to 8. The random variable Y is the number of times the spinner lands on a prime number when it is spun 12 times.

State an assumption required to model Y with a binomial distribution \text{B}(n, p).

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5b
2 marks

The random variable W is the number of times the spinner must be spun until it first lands on a '7'. The random variable L is the number of wins when the spinner is spun 10 times, where a 'win' on the first spin is scored if the spinner lands on an even number, and a 'win' on subsequent spins is scored if the spinner lands on the same number as the previous spin, or on a factor of it.

For each of W and L, give a reason why a binomial distribution would not be an appropriate model.

How did you do?

6
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1 mark

On a casino roulette wheel, the probability of the ball landing on a black number is \dfrac{9}{19}. The wheel is spun 30 times. The random variable X represents the number of times the ball lands on a black number.

Find \text{P}(X = 14).

How did you do?

7
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1 mark

As part of a marketing promotion, 47% of packets of a particular brand of crisps contain a zombie toy as a prize. A random sample of 100 packets is taken. The random variable X represents the number of packets containing a prize.

Find the value of \text{P}(X = 49).

How did you do?

8
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3 marks

On a European-style casino roulette wheel, the probability of the ball landing on a red number is \frac{18}{37}.

The wheel is spun 36 times, and the ball lands on a red number X times.

Find \text{P}(17 < X \leq 18).

How did you do?

9
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1 mark

Due to a manufacturing irregularity, 41% of Adventure Dude action figures were produced with two left hands. Although not especially rare, and therefore not especially collectible, these so-called 'double left' figures are nonetheless considered to be collector's items by hard-core Adventure Dude fanatics.

A vintage toy shop has obtained 100 Adventure Dude action figures. These may be assumed to represent a random sample.

Find the probability that exactly 45 of the 100 figures are 'double left' figures.

Give your answer to 3 significant figures.

How did you do?

10
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1 mark

On a casino roulette wheel, the probability of the ball landing on red is \dfrac{6}{13}.

The wheel is spun 50 times. The random variable X represents the number of times the ball lands on red.

Find \text{P}(X \geq 25).

Give your answer to 3 significant figures.

How did you do?

11
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2 marks

A local bakery finds that 58% of customers who buy a pastry also buy a coffee.

A random sample of 15 customers who bought a pastry is taken. The random variable X represents the number of these customers who also bought a coffee.

Find \text{P}(X = 10).

Give your answer to 3 significant figures.

How did you do?

1
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5 marks

A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, D ml, follows a normal distribution with mean 25 ml.

Given that 15% of bottles contain less than 24.63 ml, find, to 2 decimal places, the value of k such that straight P left parenthesis 24.63 less than D less than k right parenthesis equals 0.45

How did you do?

2a
2 marks

A customer service centre records every call they receive. It is found that 30% of all calls made to this centre are complaints.

A sample of 20 calls is selected. The number of calls in the sample which are complaints is denoted by the random variable X.

State two assumptions necessary for X to be modelled by a binomial distribution.

How did you do?

2b
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5 marks

Assume that X can be modelled by a binomial distribution.

(i) Find \text{P}(X = 1)

(ii) Find \text{P}(X < 4)

(iii) Find \text{P}(X \geq 10)

How did you do?

2c
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3 marks

In a random sample of 10 calls to a school, the number of calls which are complaints, Y, may be modelled by a binomial distribution Y \sim \text{B}(10 , \; p).

The standard deviation of Y is 1 . 5

Calculate the possible values of p.

How did you do?

3a
2 marks

Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick.

Every time he attempts the trick, there is a probability of 0 . 2 that he will fall off his skateboard.

Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

(i) Find the mean number of times he falls off in a day.

(ii) Find the variance of the number of times he falls off in a day.

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3b
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5 marks

(i) Find the probability that, on a particular day, he falls off exactly 10 times.

(ii) Find the probability that, on a particular day, he falls off 5 or more times.

How did you do?

3c
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3 marks

Patrick has 30 attempts to perform the trick on each of 5 consecutive days.

(i) Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.

(ii) Explain why it may be unrealistic to use the same value of 0 . 2 for the probability of falling off for all 5 days.

How did you do?

4a
1 mark

Abu visits his local hardware store to buy six light bulbs. He knows that 15% of all bulbs at this store are faulty.

State a distribution which can be used to model the number of faulty bulbs he buys.

How did you do?

4b
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1 mark

Find the probability that all of the bulbs he buys are faulty.

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4c
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2 marks

Find the probability that at least two of the bulbs he buys are faulty.

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4d
1 mark

Find the mean of the distribution stated in part (a).

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4e
2 marks

State two necessary assumptions in context so that the distribution stated in part (a) is valid.

How did you do?

5a
2 marks

For each of the following, state with a reason whether the random variable is discrete or continuous.

(i) A student cuts a 1-metre length of rope into two pieces at a random point. The random variable A is the length of the shorter piece.

(ii) You survey a sample of students about their preferences for after-school activities. The random variable B is the number of students who prefer lawn bowling.

How did you do?

5b
3 marks

Three histograms show the distributions of the random variables D, E and F.

Three histograms: (i) bell-shaped for Measurements of D, (ii) bimodal for Measurements of E, (iii) increasing trend for Measurements of F.

For each of D, E and F, state with a reason whether a normal distribution would be an appropriate model.

How did you do?

1a
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3 marks

A survey of 120 adults found that the volume, X litres per person, of carbonated drinks they consumed in a week had the following results:

\sum x = 165 . 6

\sum x^{2} = 261 . 8

(i) Calculate the mean of X.

(ii) Calculate the standard deviation of X.

How did you do?

1b
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3 marks

Assuming that X can be modelled by a normal distribution, find

(i) \text{P}(0 . 5 < X < 1 . 5)

(ii) \text{P}(X = 1)

How did you do?

1c
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2 marks

Determine, with a reason, whether a normal distribution is suitable to model this data.

How did you do?

1d
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4 marks

It is known that the volume, Y litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with mean \mu and standard deviation 0 . 21

Given that \text{P}(Y > 0 . 75) = 0 . 10, find the value of \mu, correct to three significant figures.

How did you do?

2a
1 mark

It is given that

X \sim \text{B}(48 , 0 . 175)

Find the mean of X.

How did you do?

2b
1 mark

Show that the variance of X is 6 . 93

How did you do?

2c
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1 mark

Find \text{P}(X < 10)

How did you do?

2d
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2 marks

Find \text{P}(X \geq 6)

How did you do?

2e
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2 marks

Find \text{P}(9 \leq X \leq 15)

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2f
2 marks

The aeroplanes used on a particular route have 48 seats.

The proportion of passengers who use this route to travel for business is known to be 17.5%

Make two comments on whether it would be appropriate to use X to model the number of passengers on an aeroplane who are travelling for business using this route.

How did you do?

3a
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3 marks

In a particular year, the height of a male athlete at the Summer Olympics has a mean 1 . 78 metres and standard deviation 0 . 23 metres.

The heights of 95% of male athletes are between 1 . 33 metres and 2 . 22 metres.

Comment on whether a normal distribution may be suitable to model the height of a male athlete at the Summer Olympics in this particular year.

How did you do?

3b
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3 marks

You may assume that the height of a male athlete at the Summer Olympics may be modelled by a normal distribution with mean 1 . 78 metres and standard deviation 0 . 23 metres.

(i) Find the probability that the height of a randomly selected male athlete is 1 . 82 metres.

(ii) Find the probability that the height of a randomly selected male athlete is between 1 . 70 metres and 1 . 90 metres.

(iii) Two male athletes are chosen at random. Calculate the probability that both of their heights are between 1 . 70 metres and 1 . 90 metres.

How did you do?

1a
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2 marks

A medical researcher is studying the number of hours, T, a patient stays in hospital following a particular operation.

The histogram summarises the results for a random sample of 90 patients.

Histogram showing frequency density of time in hours. 0-4 bar has height 2.5, 4-7 bar has height 3, 7-12 bar has height 4.2, 12-16 bar has height 4, 16-20 bar has height 3.5, 20-40 bar has height 1.

Use the histogram to estimate straight P open parentheses 10 less than T less than 30 close parentheses.

How did you do?

1b
1 mark

For these 90 patients the time spent in hospital following the operation had

  • a mean of 14.9 hours

  • a standard deviation of 9.3 hours

Tomas suggests that T can be modelled by straight N open parentheses 14.9 comma space 9.3 squared close parentheses

With reference to the histogram, state, giving a reason, whether or not Tomas’ model could be suitable.

How did you do?

1c
4 marks

Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation

y equals k x straight e to the power of negative x end exponent space space space 0 less or equal than x less or equal than 4

where

  • x is measured in tens of hours

  • k is a constant

Use algebraic integration to show that

integral subscript 0 superscript n x straight e to the power of negative x end exponent d x equals 1 minus open parentheses n plus 1 close parentheses straight e to the power of negative n end exponent

How did you do?

1d
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3 marks

Show that, for Xiang’s model, k equals 99 to the nearest integer.

How did you do?

1e
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1 mark

Estimate straight P open parentheses 10 less than T less than 30 close parentheses using Tomas' model of T space tilde space straight N open parentheses 14.9 comma space 9.3 squared close parentheses.

How did you do?

1f
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2 marks

Estimate straight P open parentheses 10 less than T less than 30 close parentheses using Xiang’s curve with equation y equals 99 x straight e to the power of negative x end exponent and the answer to part (c).

How did you do?

1g
1 mark

The researcher decides to use Xiang’s curve to model straight P open parentheses a less than T less than b close parentheses.

State one limitation of Xiang’s model.

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