Choosing Distributions (A Level only) (AQA A Level Maths: Statistics)

State the conditions that must be satisfied to be able to model a random variable  with a binomial distribution .

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.

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

The random variable 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.

The random variable 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.

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 is the number of losses when the spinner is spun and the coin is flipped twelve times.

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

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.

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

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

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

Each of the following histograms shows the distribution of results when a large number of measurements of the random variables or  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.

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.

For each of the following binomial random variables, :

• if  can be approximated by a normal distribution, then write down the normal approximation to  in the form , giving the values of  and 
• if  cannot be approximated by a normal distribution, then give a reason why

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution .

On a casino roulette wheel, the probability of the ball landing on a black number is .

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

Find .

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

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

State the conditions that must be satisfied to be able to model a random variable  with a binomial distribution .

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.

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

The random variable 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.

The random variable 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.

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 is the number of losses when the spinner is spun and the coin is flipped twelve times.

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

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.

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

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

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

Each of the following histograms shows the distribution of results when a large number of measurements of the random variables or  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.