OCR GCE AS Level Further Maths B (MEI) Statistics a (optional - choose any two of Y411/Y412/Y413/Y414) June 2025 question paper Y412

A student is investigating what level of exercise is given to dogs in the UK by their owners.

When trying to collect suitable information, give two reasons why it might be advantageous

to use a sample of dog owners, rather than a census of dog owners.

Explain why it is preferable that the size of the sample is large.

The probability distribution of a discrete random variable X is given in the table.

Find the value of a.

The probability function of another discrete random variable, Y, is defined below.

where n is a positive integer.

Determine the value of n.

Find P (Y < n).

Given that X and Y are independent, determine P (X + Y ⩽ 2).

The discrete random variable Q has the distribution Geo(0.3).

Find P (Q = 2).

Determine P (Q ⩽ 5).

The discrete random variable R has a uniform distribution on {4, 5,..., 8}.

The discrete random variable S has the distribution B (10, 0.1).

Q, R and S are mutually independent of each other.

Find P (Q + R + S = 5)

The discrete random variable T is defined by T = R - Q + 2 S

Determine the value of E (T).

Give two reasons why it would be appropriate for a hypothesis test based on the

product‑moment correlation coefficient to be carried out in this situation.

The scientist analyses the collected data and produces the following summary statistics.

Find and use the equation of a suitable regression line to determine an estimate of the length

of an adult female badger of mass 10 kg.

Two judges, Judge A and Judge B, independently rank 8 entries in a jam‑making competition, awarding each jam a rank from 1 (best) to 8 (worst). There are no tied ranks. The ranks are shown in the table below but unfortunately two of the rankings awarded by Judge B have been lost.

Explain which jam is likely to be the best.

Carry out a hypothesis test to make the assessment using a significance level of 1%.

The number of misprints in every 100 000 words of a book is denoted by the random variable X. The probability of any given word being misprinted is 0.000 11. You may assume that misprints occur randomly and independently of each other.

Explain why a certain binomial distribution could be used to model the distribution of X.

Hence explain why X could also be modelled by a certain Poisson distribution.

State the mean of this Poisson distribution.

For the remainder of this question, you should use a Poisson distribution to model the number of misprints in a book.

For a book containing exactly 100 000 words, find the probability that it contains exactly 9

misprints.

A proof‑reader is paid £20 for each misprint identified in this book. You may assume that the proof‑reader identifies every misprint in the book. The payment in pounds made to the proof‑reader is denoted by the random variable P.

By considering the mean and variance of P, show that P does not follow a Poisson

distribution.

The proof‑reader is paid for proof‑reading another book containing exactly 50 000 words, again identifying every misprint in the book.

Determine the probability that the proof‑reader is paid a total amount which is more than

£300 and less than £500 for proof‑reading both books.

The manager of a Health Service Trust receives data from two hospitals, Hospital A and Hospital B, about the lengths of stays in hospital for patients with a particular condition. The data are in the form of a list of the number of days that each patient with that condition has spent at each hospital in one particular year.

The manager decides to categorise the length of each stay according to Table 7.1 before putting the data into a contingency table.

Table 7.1

State one advantage and one disadvantage of categorising the data in this way.

The contingency table created by the manager is shown in Table 7.2.

Table 7.2

The manager wishes to use a chi‑squared test to assess whether there is any association between the length of stay of the patient at the hospital and which hospital is treating the patient. You may assume that the lengths of stay are independent of each other and the data is a random sample from the underlying population of patients with this condition.

Write down suitable null and alternative hypotheses for the test.

Table 7.3 shows some of the expected frequencies. The table is incomplete.

Complete the copy of Table 7.3 in the Printed Answer Booklet. Your entries should be

given to 3 decimal places.

Table 7.4 shows some of the chi‑squared contributions. Entries are given to 4 decimal places. The table is incomplete.

Complete the copy of Table 7.4 in the Printed Answer Booklet and carry out the test at the

10% significance level. You do not need to restate your hypotheses.

Use the chi‑squared contributions found in part (d) to explain which length stay, at which

hospital, gives most evidence of any association.

The manager receives a new instruction that all hypothesis tests carried out by the trust should actually have a significance level of 5%.

Explain whether following this new instruction would have affected the result of the test.