AQA A Level Further Maths Paper 3S June 2022 question paper 7367/3S

The random variable T follows a discrete uniform distribution and can take values

1, 2, 3, …, 16

Find the variance of T

Circle your answer.

1.25 18.75 21.25 21.33

The random variable X has probability density function

Circle your answer.

The random variable X has an exponential distribution with probability density function

Given that 2, find P(X 1), giving your answer to three decimal places.

Daisies and dandelions are the only flowers growing in a field.

The number of daisies per square metre in the field has a mean of 16

The number of dandelions per square metre in the field has a mean of 10

The number of daisies per square metre and the number of dandelions per square metre are independent.

Using a Poisson model, find the probability that a randomly selected square metre from the field has a total of at least 30 flowers, giving your answer to three decimal places.

A survey of the entire field is taken.

The standard deviation of the total number of flowers per square metre is 10

State, with a reason, whether the model used in part (a) is valid.

A 99% confidence interval for the population mean of the masses of the apples is constructed using the random sample.

Show that the confidence interval is (81.7, 91.3) with values correct to three significant figures.

Padraig claims that the population mean mass of the apples is 85 grams.

He carries out a hypothesis test at the 1% level of significance using the random sample of 12 apples.

The hypotheses are

State, with a reason, whether the null hypothesis is accepted or rejected.

Interpret, in context, the conclusion to the hypothesis test in part (b).

The discrete random variable X has probability distribution function

where a, b and c are constants.

The mean of X is 1.2 and the variance of X is 0.56

Deduce the values of a, b and c

The continuous random variable Y is independent of X and has variance 15

A scientist is investigating the air quality in two countries, A and B

Measurements of the air quality are taken at sites throughout both countries and assigned a rank.

The air quality is given Rank 1 if the level of air pollution is lower than a specified threshold.

The air quality is given Rank 2 if the level of air pollution is higher than the specified threshold.

A random sample of 500 measurements is collected.

The results are summarised in the following table.

The scientist claims that there is an association between country and air quality.

Test the scientist’s claim, using the 10% level of significance.

For the context of the test carried out in part (a), state the meaning of a Type I error.

The continuous random variable X has cumulative distribution function F (x) where

Show that the median of X is c, where a, b and c are integers to be found.

Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12

The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to p

Lianne will only buy the battery if p is more than 0.4

Determine whether Lianne will buy the battery.

A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged.

Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges.