A LevelMathsEdexcelStatisticsExam QuestionsStatistical DistributionsWorking with Distributions

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

Exam code: 9MA0

3 hours23 questions
Easy
Medium
Hard
Very Hard
1a
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.

How did you do?

1b
1 mark

Write down the name of the probability distribution of A, the random variable described in the first row of the table in part (a).

How did you do?

2a
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.

How did you do?

2b
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.

How did you do?

2c
1 mark

State which one of the random variables A, B and C from part (b) follows a discrete uniform distribution.

    Choose your answer

    3a
    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.

    How did you do?

    3b
    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.

    How did you do?

    4a
    1 mark

    The random variable X tilde text B end text open parentheses n comma close open p close parentheses can be approximated by Y tilde text N end text open parentheses mu comma close open sigma squared close parentheses when certain conditions are met.

    State the condition on n that is required for this approximation to be valid.

    How did you do?

    4b
    2 marks

    (i) State the value of p for which the normal approximation to a binomial distribution is most accurate.

    (ii) Give a reason to support your answer.

    How did you do?

    4c
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    4 marks

    For each of the following binomial random variables X, state with reasons whether X can be approximated by a normal distribution. Where it can, write down the normal approximation in the form text N end text open parentheses mu close comma open sigma squared close parentheses.

    (i) X tilde text B end text open parentheses 6 comma close open 0.45 close parentheses

    (ii) X tilde text B end text open parentheses 60 comma close open 0.05 close parentheses

    (iii) X tilde text B end text open parentheses 60 comma close open 0.45 close parentheses

    How did you do?

    5a
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    2 marks

    The random variable X tilde text B end text open parentheses 100 comma close open 0.36 close parentheses can be approximated by Y tilde text N end text open parentheses mu comma close open sigma squared close parentheses.

    Find the value of \mu and show that \sigma = 4 . 8.

    How did you do?

    5b
    1 mark

    Explain why a continuity correction must be applied when using this approximation.

    How did you do?

    5c
    4 marks

    Using a continuity correction, find the value of k in each of the following:

    (i) \text{P}(X \leq 30) \approx \text{P}(Y < k)

    (ii) \text{P}(X < 30) \approx \text{P}(Y < k)

    (iii) \text{P}(X \geq 30) \approx \text{P}(Y > k)

    (iv) \text{P}(X > 30) \approx \text{P}(Y > k)

    How did you do?

    6a
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    4 marks

    The random variable X \sim \text{B}(150, 0.6) is to be approximated by Y \sim \text{N}(\mu, \sigma^{2}).

    (i) State two reasons why a normal approximation is suitable.

    (ii) Find the values of \mu and \sigma.

    How did you do?

    6b
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    3 marks

    Use a normal approximation to estimate \text{P}(87 \leq X \leq 101). Clearly state the continuity correction used.

    How did you do?

    1a
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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

    How did you do?

    1b
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    3 marks

    George now throws the ball at the target 250 times.

    Use a normal approximation to calculate the probability that he will hit the target more than 110 times.

    How did you do?

    2a
    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).

    How did you do?

    2b
    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?

    2c
    1 mark

    The random variable S is the number shown when the spinner is spun once. Name the probability distribution that would be appropriate for S.

    How did you do?

    3a
    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?

    3b
    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?

    4
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    3 marks

    For each of the following binomial random variables X:

    • state, with reasons, whether X can be approximated by a normal distribution

    • if appropriate, write down the normal approximation to X in the form \text{N}(\mu, \sigma^{2}), giving the values of \mu and \sigma^{2}.

    (i) X \sim \text{B}(10, 0.5)

    (ii) X \sim \text{B}(80, 0.54)

    How did you do?

    5a
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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?

    5b
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    3 marks

    In a separate experiment, the wheel is spun 1000 times. The random variable Y represents the number of times the ball lands on a black number.

    (i) State, giving reasons, why a normal approximation is appropriate for Y.

    (ii) Write down the normal distribution that could be used to approximate Y.

    How did you do?

    5c
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    2 marks

    Use the distribution from part (b)(ii) to estimate the probability that the ball lands on a black number at least 500 times in the 1000 spins.

    How did you do?

    6a
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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?

    6b
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    1 mark

    Write down the normal distribution that could be used to approximate X.

    How did you do?

    6c
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    2 marks

    Using the normal distribution from part (b), estimate \text{P}(X = 49). You must state the continuity correction used.

    How did you do?

    1a
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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?

    1b
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    3 marks

    A random sample of 200 bottles is taken.

    Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and k ml.

    How did you do?

    2a
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    2 marks

    A type of seed has a germination rate of 55% when planted under standard conditions. A researcher plants 60 seeds and the random variable Y represents the number that germinate.

    Write down the normal distribution that can be used to approximate Y, giving the values of \mu and \sigma^2. State two reasons why this approximation is valid.

    How did you do?

    2b
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    3 marks

    Use your normal approximation to estimate the probability that at least 40 seeds germinate.

    How did you do?

    3a
    2 marks

    A factory quality control team monitors a production line. Records show that 60% of components pass the quality check on the first attempt. In a sample of 1200 components, the random variable W is the number of components that pass on the first attempt.

    Explain why W can be approximated by a normal distribution. Give your reasons in the context of the problem.

    How did you do?

    3b
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    3 marks

    Use a normal approximation to estimate the probability that fewer than 700 components pass the quality check on the first attempt, applying a continuity correction.

    How did you do?

    4a
    2 marks

    The random variable W \sim \text{B}(500, 0.4).

    Give two reasons why a normal distribution can be used to approximate W.

    How did you do?

    4b
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    6 marks

    Find, using the normal approximation:

    (i) \text{P}(189 \leq W \leq 211)

    (ii) \text{P}(W > 220)

    How did you do?

    4c
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    3 marks

    Using the normal approximation, find the largest integer value of w such that \text{P}(W \leq w) < 0.1.

    How did you do?

    5a
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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?

    5b
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    4 marks

    In a separate experiment, the wheel is spun 1000 times. Let Y be the number of times the ball lands on a red number.

    Use a normal approximation to estimate the probability that the ball lands on red more than half of the time. Give your answer to 3 significant figures.

    How did you do?

    6a
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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?

    6b
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    3 marks

    Use a normal approximation to estimate the probability that exactly 45 of the 100 figures are 'double left' figures.

    Give your answer to 3 significant figures.

    How did you do?

    6c
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    3 marks

    Using the normal approximation from part (b), find the probability that fewer than 40 of the 100 figures are 'double left' figures.

    Give your answer to 3 significant figures.

    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
    Sme Calculator
    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.

    How did you do?

    2a
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    2 marks

    A chocolate factory produces boxes of chocolates. 44% of the chocolates produced contain a caramel centre.

    A box contains 80 chocolates. The random variable X represents the number of chocolates in the box with a caramel centre.

    State, giving a reason, a suitable normal distribution that can be used to approximate X. Give your answer in the form \text{N}(\mu, \sigma^2).

    How did you do?

    2b
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    4 marks

    Using your approximation from part (a), estimate the probability that at least 40 of the 80 chocolates in the box contain a caramel centre.

    Give your answer to 3 significant figures.

    How did you do?

    2c
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    5 marks

    The same factory also produces boxes of premium chocolates. Only 2% of these premium chocolates contain a rare pistachio filling.

    A box of premium chocolates contains 100 chocolates. The random variable Y represents the number of these chocolates with a pistachio filling. A quality controller suggests that Y can be approximated by a normal distribution with mean np and variance np(1-p).

    (i) Write down this proposed normal approximation for Y, in the form \text{N}(\mu, \sigma^2).

    (ii) By calculating \text{P}(Y < 0) for this proposed approximation, explain why it is a poor model for the number of premium chocolates containing a pistachio filling.

    How did you do?

    3a
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    2 marks

    A manufacturer produces ceramic tiles. Historically, 40% of the tiles produced have a slight colour variation.

    A random sample of 980 tiles is selected. The random variable W represents the number of tiles in the sample with a colour variation.

    State, giving reasons, why the distribution of W can be approximated by a normal distribution.

    How did you do?

    3b
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    4 marks

    Use a normal approximation to estimate the probability that at least 387 and no more than 397 of the tiles in the sample have a colour variation.

    Give your answer to 3 significant figures.

    How did you do?

    3c
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    2 marks

    Use the normal approximation to estimate the probability that at least 400 of the tiles in the sample have a colour variation.

    Give your answer to 3 significant figures.

    How did you do?

    3d
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    3 marks

    Using the normal approximation, find the largest integer w such that the probability of obtaining fewer than w tiles with a colour variation is less than 0.05.

    How did you do?

    4a
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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?

    4b
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    2 marks

    The wheel is now spun 1000 times. The random variable Y represents the number of times the ball lands on red.

    State, giving reasons, why the distribution of Y can be approximated by a normal distribution.

    How did you do?

    4c
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    4 marks

    Use a normal approximation to estimate the probability that in 1000 spins the ball lands on red at least 450 times but no more than 475 times.

    Give your answer to 3 significant figures.

    How did you do?

    5a
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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?

    5b
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    2 marks

    On a busy Saturday, a random sample of 150 customers who bought a pastry is taken. The random variable Y represents the number of these customers who also bought a coffee.

    State, giving reasons, why the distribution of Y can be approximated by a normal distribution.

    How did you do?

    5c
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    4 marks

    Use a normal approximation to estimate the probability that between 80 and 90 (inclusive) of the 150 customers in the sample also bought a coffee.

    Give your answer to 3 significant figures.

    How did you do?

    5d
    1 mark

    A student attempts to use the normal approximation from part (c) to estimate the probability that more than 150 of the customers in the sample bought a coffee.

    Explain why this is not a sensible use of the normal approximation.

    How did you do?