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

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 $\mathrm{P}(24.63<D<k)=0.45$

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.

Assume that $X$ can be modelled by a binomial distribution.

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

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

(iii) Find $\text{P}(X\ge 10)$

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 $Y~\text{B}(10,p)$.

The standard deviation of $Y$ is $1.5$

Calculate the possible values of $p$.

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.

(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.

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.

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.

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

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

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

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

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.

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

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

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 $X$.

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

(i) $\text{P}(0.5<X<1.5)$

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

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

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.

It is given that

$X~\text{B}(48,0.175)$

Find the mean of $X$.

Show that the variance of $X$ is $6.93$

Find $\text{P}(X<10)$

Find $\text{P}(X\ge 6)$

Find $\text{P}(9\le X\le 15)$

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.

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.

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.

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.

Use the histogram to estimate $\mathrm{P}\left(10<T<30\right)$.

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 $\mathrm{N}\left(14.9,9.3^2\right)$

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

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

$y=kx{\mathrm{e}}^{-x}0\le x\le 4$

where

• $x$ is measured in tens of hours

• $k$ is a constant

Use algebraic integration to show that

${\int }_0^{n}x{\mathrm{e}}^{-x}dx=1-\left(n+1\right){\mathrm{e}}^{-n}$

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

Estimate $\mathrm{P}\left(10<T<30\right)$ using Tomas' model of $T~\mathrm{N}\left(14.9,9.3^2\right)$.

Estimate $\mathrm{P}\left(10<T<30\right)$ using Xiang’s curve with equation $y=99x{\mathrm{e}}^{-x}$ and the answer to part (c).

The researcher decides to use Xiang’s curve to model $\mathrm{P}\left(a<T<b\right)$.

State one limitation of Xiang’s model.