A LevelMathsOCRMaths AStatisticsExam QuestionsStatistical DistributionsChoosing Distributions

Choosing Distributions (OCR A Level Maths A: Statistics): Exam Questions

Exam code: H240

1 hour10 questions
Easy
Medium
Hard
1a
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The table below shows five scenarios involving different random variables. Complete the table by placing a cross (×) in the correct box to indicate whether the random variable can be modelled by a binomial distribution, a normal distribution or neither. The first scenario is 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 of the nine counters. The random variable A represents the number that is written on the selected counter.

 

 

cross times

A farmer has many hens. The random variable B represents the mass 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 break for lunch. 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 dice with sides labelled 1 to 6. The random variable E represents the number of students who roll a number less than 5.

 

 

 

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1b
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Write down the name of the probability distribution of A, the random variable described in part (a).

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2a
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In an experiment there are a fixed number of trials and each trial results in a success or failure. Let X be the number of successful trials. Write down the two other conditions that would need to be present to make X follow a binomial distribution.

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2b
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A fair spinner has 8 sectors labelled with the numbers 1 through 8. For each of the following cases, give a reason to explain why a binomial distribution would not be appropriate for modelling the specified random variable.

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

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

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

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2c
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State which one of the random variables defined in part (b) follows a discrete uniform distribution.

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3a
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For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

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

(ii) 100 students sit a test which is marked out of 50. The random variable B is the number of marks achieved 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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3b
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The following histogram shows the distribution of results when a large number of measurements of the specified random variable D are made. State with a reason whether a normal distribution would be appropriate for modelling the random variable.

q3b-4-4-easy-aqa-a-level-maths-statistics

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4a
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The random variable X tilde space B left parenthesis n comma space p right parenthesis can be approximated by Y tilde N left parenthesis mu comma space sigma squared right parenthesis when certain conditions are fulfilled.

State the condition for n which is required to use this approximation.

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4b
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(i) State the value of p that will give the most accurate estimate.

(ii) Give a reason to support your value.

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4c
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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 N left parenthesis mu comma space sigma squared right parenthesis, giving the values of mu and sigma squared.

(i) X tilde B left parenthesis 6 comma space 0.45 right parenthesis

(ii) X tilde B left parenthesis 60 comma space 0.05 right parenthesis

(iii) X tilde B left parenthesis 60 comma space 0.45 right parenthesis

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1a
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State the conditions that must be satisfied to be able to model a random variable X with a binomial distribution B(n comma space p).

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1b
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A fair spinner has 5 sectors labelled with the numbers 1 through 5. The spinner is spun and a fair coin is flipped, and the number the spinner lands on along with the result of the coin flip (heads or tails) are recorded. For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

(i) If the coin lands on heads, then the random variable S is the number of the sector that the spinner lands on times two. Otherwise S is the number of the sector that the spinner lands on plus 10.

(ii) The random variable W is the number of times the spinner is spun and the coin is flipped until an odd number on the spinner occurs together with tails on the coin.

(iii) The random variable Y is the number of times a prime number on the spinner occurs together with heads on the coin, when the spinner is spun and the coin is flipped 21 times.

(iv) Each time the spinner is spun and the coin is flipped, it is a 'win' if a square number on the spinner occurs together with heads on the coin, or it is a 'loss' if a non-square number on the spinner occurs together with tails on the coin. Any other outcome is a 'draw'. The random variable L is the number of losses when the spinner is spun and the coin is flipped twelve times.

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1c
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For the random variable S defined in (b) (i) above, give the name of the probability distribution that would be appropriate for modelling S. Justify your answer.

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2a
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For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i) A cake recipe calls for a certain amount of flour to be used. The random variable A is the number of cakes that can be made, following the recipe exactly each time, from a bag containing a random amount of flour.

(ii) A student cuts a one-metre length of rope into two pieces at a random point. The random variable B is the difference in length between the two pieces of rope that result.

(iii) People are chosen at random from the UK population. The random variable C is the age of a randomly selected person, measured from their date and time of birth.

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2b
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Each of the following histograms shows the distribution of results when a large number of measurements of the random variables D comma space E or F are made. In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable. Where a normal model is appropriate, suggest a real-world variable that might show such a distribution.

q2b-4-4-easy-aqa-a-level-maths-statistics

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3a
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State the conditions that must be met for the distribution of a binomial random variable to be able to be approximated by a normal random variable.

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3b
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For each of the following binomial random variables, X tilde B left parenthesis n comma space p right parenthesis:

  • if X can be approximated by a normal distribution, then write down the normal approximation to X in the form N left parenthesis mu comma space sigma squared right parenthesis, giving the values of mu and sigma

  • if X cannot be approximated by a normal distribution, then give a reason why

(i) X tilde B left parenthesis 15 comma space 0.5 right parenthesis

(ii) X tilde B left parenthesis 150 comma space 0.56 right parenthesis

(iii) X tilde B left parenthesis 1500 comma space 0.005 right parenthesis

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4a
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Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution B left parenthesis n comma space p right parenthesis.

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4b
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On a casino roulette wheel, the probability of the ball landing on a black number is 9 over 19.

The wheel is spun 30 times, and the ball lands on a black number X times.

Find straight P open parentheses X equals 14 close parentheses.

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4c
Sme Calculator3 marks

In a separate experiment, the wheel is spun 1000 times and Y, the number of times the ball lands on a black number, is recorded.

(i) Explain why a normal approximation would be appropriate in this case.

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

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1a
Sme Calculator4 marks

State the conditions that must be satisfied to be able to model a random variable X with a binomial distribution B left parenthesis n comma space p right parenthesis.

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1b
Sme Calculator4 marks

A fair spinner has 5 sectors labelled with the numbers 1 through 5. The spinner is spun and a fair coin is flipped, and the number the spinner lands on along with the result of the coin flip (heads or tails) are recorded. For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

(i) If the coin lands on heads, then the random variable S is the number of the sector that the spinner lands on times two. Otherwise S is the number of the sector that the spinner lands on plus 10.

(ii) The random variable W is the number of times the spinner is spun and the coin is flipped until an odd number on the spinner occurs together with tails on the coin.

(iii) The random variable Y is the number of times a prime number on the spinner occurs together with heads on the coin, when the spinner is spun and the coin is flipped 21 times.

(iv) Each time the spinner is spun and the coin is flipped, it is a 'win' if a square number on the spinner occurs together with heads on the coin, or it is a 'loss' if a non-square number on the spinner occurs together with tails on the coin. Any other outcome is a 'draw'. The random variable L is the number of losses when the spinner is spun and the coin is flipped twelve times.

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1c
Sme Calculator2 marks

For the random variable S defined in (b) (i) above, give the name of the probability distribution that would be appropriate for modelling S. Justify your answer.

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2a
Sme Calculator3 marks

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

(i) A cake recipe calls for a certain amount of flour to be used. The random variable A is the number of cakes that can be made, following the recipe exactly each time, from a bag containing a random amount of flour.

(ii) A student cuts a one-metre length of rope into two pieces at a random point. The random variable B is the difference in length between the two pieces of rope that result.

(iii) People are chosen at random from the UK population. The random variable C is the age of a randomly selected person, measured from their date and time of birth.

How did you do?

2b
Sme Calculator4 marks

Each of the following histograms shows the distribution of results when a large number of measurements of the random variables D comma space E or F are made. In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable. Where a normal model is appropriate, suggest a real-world variable that might show such a distribution.

q2b-4-4-easy-aqa-a-level-maths-statistics

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