OCR A Level Further Maths B (MEI) Statistics Minor (minor option - choose one with a major, or three alone) June 2025 question paper Y432

A discrete random variable, X, has a Poisson distribution with mean 4.

Write down the value of Var (X).

Find P (X = 3).

Find P (X > 3).

Another discrete random variable, Y, which is independent of X, has a Poisson distribution with mean 5.

Find P (X + Y ⩽ 9).

By first determining the equation of an appropriate regression line, estimate the value of v

when t = 45.

Comment on the reliability of your estimate in part (a).

Explain why in this scenario it would be inappropriate to use a value of v to estimate a value

of t using any regression line.

An archer fires arrows one by one at a target until an arrow hits the target. The number of arrows fired up to and including the arrow that hits the target is denoted by X. The archer considers using a geometric model for the distribution of X.

State two assumptions required for such a model to be appropriate.

For the remainder of this question you may assume that a geometric model is appropriate for the distribution of X.

The archer keeps records of the values of X. He collects a sample of these values and uses them to estimate the variance of X.

Give two features that the sample should have.

The estimate of the variance of X calculated from the sample is 12.

Determine an estimate of the probability that when the archer fires a single arrow at the

target, the arrow hits the target.

Use your answer to part (c) to write down an estimate of the expected number of arrows that

the archer must fire at the target up to and including the first arrow that hits the target.

The discrete random variable, X, has a uniform distribution over the values {5, 6,..., n }.

In this question you must show detailed reasoning.

In part (a) you should assume that n = 24.

Use the substitution Y = X - 4 and the given formula in the formulae booklet to

determine Var (X).

A student states that P(X > E(X)) = for all n ⩾ 24.

Explain whether the student is correct.

Long-term data collected at a certain farm indicates that 30% of eggs that it produces are large and the rest are categorised as medium.

Eggs are collected in batches of 4 and the number of large eggs in each batch, X, is recorded.

It is suggested that a binomial distribution might be appropriate to model X.

State two reasons, within the context of this question, why a binomial model might be

appropriate to model X.

State two assumptions, within the context of this question, that must be made for a binomial model to be appropriate to model X. You should not refer to the reasons given

in part (a)(i).

Write down the binomial distribution which might be a suitable model for the

distribution of X.

The farm manager investigates whether the distribution in part (a)(iii) is a good model for X.

State one reason why taking data from a sample of 6 batches would not be appropriate to

carry out the investigation.

The farm manager takes a random sample of 100 batches. The results are summarised in Table 5.1.

Table 5.1

Table 5.2, shows expected frequencies for the values of X in 100 batches, assuming that X has the distribution from part (a)(iii). The table is incomplete.

Table 5.2

Complete the copy of Table 5.2 in the Printed Answer Booklet. Give the expected

frequencies to 2 decimal places.

Carry out a χ² test, at the 5% significance level, for goodness of fit of the model in

part (a)(iii).

A factory discharges very low concentrations of an inert chemical through an outfall into a nearby body of water.

The concentration of the chemical is measured by sensors at seven known locations at fixed distances from the outfall. Each measurement is known to be affected by various environmental factors. None of these factors are measured or noted.

The manager of the factory needs to assess the relationship between the concentration of the chemical at each sensor and the distance of the sensor from the outfall.

The manager can carry out a hypothesis test either for any correlation or for any association between the concentration of the chemical at the sensor and the distance of the sensor from the outfall.

Explain why the manager should carry out a test for association rather than one for

correlation.

The concentration of the chemical at each sensor and the distance of each sensor from the outfall, both in arbitrary units, for a particular day after the chemical has been discharged are given in the table below.

In this question you must show detailed reasoning.

Carry out the appropriate test for association at the 5% significance level. A copy of the table, with spare, empty rows, is given in the Printed Answer Booklet to help with any calculations.

By forming an expression for E (W) from the table found in (c) and then eliminating β, show

that E (W) = 2E (X).

In the case where α = 0.1 and β = 0.7, find P(W = 0|X₁ = 1).