OCR A Level Further Maths B (MEI) Statistics Major (major option - choose one of Y421/Y422, or three minors) November 2021 question paper Y422

When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm. The sample standard deviation head circumference is 1.530 cm.

Determine a 95% confidence interval for the population mean head circumference of newborn

female babies.

Explain why you can calculate this interval even though the distribution of the population of

head circumferences of newborn female babies is unknown.

In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable X represents the difference between the highest and lowest scores.

The table shows the probability distribution of X.

Draw a graph to illustrate the distribution.

Describe the shape of the distribution.

In this question you must show detailed reasoning.

Find each of the following.

• E(X)

• Var(X)

As a result of playing the game, the player receives 30 X pence from the organiser of the game.

Find the variance of the amount that the player receives.

The player pays k pence to play the game.

Given that the average profit made by the organiser is 12.5 pence per game, determine the

value of k.

In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly.

At a particular site, on average 4% of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.

Find the probability that in a sequence of messages, the 10th message is the first one which is

misunderstood.

Find the probability that in a sequence of 20 messages, there are no misunderstood messages.

Determine the expected number of messages required for 3 of them to be misunderstood.

Determine the probability that in a sequence of messages, the 3rd misunderstood message is

the 60th message in the sequence.

A radioactive source contains 1 000 000 nuclei of a particular radioisotope. On average 1 in 200 000 of these nuclei will decay in a period of 1 second. The random variable X represents the number of nuclei which decay in a period of 1 second. You should assume that nuclei decay randomly and independently of each other.

Explain why you could use either a binomial distribution or a Poisson distribution to model

the distribution of X.

Use a Poisson distribution to answer parts (b) and (c).

Calculate each of the following probabilities.

• P(X = 6)

• P(X > 6)

Determine an estimate of the probability that at least 60 nuclei decay in a period of 10 seconds.

A manufacturer uses three types of capacitor in a particular electronic device. The capacitances, measured in suitable units, are modelled by independent Normal distributions with means and standard deviations as shown in the table.

Determine the probability that the total capacitance of a randomly chosen capacitor of Type B

and two randomly chosen capacitors of Type A is at least 16 units.

Determine the probability that the capacitance of a randomly chosen capacitor of Type C is

within 1 unit of the total capacitance of four randomly chosen capacitors of Type B.

When the manufacturer gets a new batch of 1000 capacitors from the supplier, a random sample of 10 of them is tested to check the capacitances. For a new batch of Type C capacitors, summary statistics for the capacitances, x units, of the random sample are as follows.

You should assume that the capacitances of the sample come from a Normally distributed population, but you should not assume that the standard deviation is 0.64 as for previous Type C capacitors.

In this question you must show detailed reasoning.

Carry out a hypothesis test at the 5% significance level to check whether it is reasonable to assume that the capacitors in this batch have the specified mean capacitance for Type C of

30.2 units.

Cosmic rays passing through the upper atmosphere cause muons, and other types of particle, to be formed. Muons can be detected when they reach the surface of the earth. It is known that the mean number of muons reaching a particular detector is 1.7 per second. The numbers of muons reaching this detector in 200 randomly selected periods of 1 second are shown in Fig. 6.1.

Use the values of the sample mean and sample variance to discuss the suitability of a Poisson

distribution as a model.

The screenshot in Fig. 6.2 shows part of a spreadsheet to assess the goodness of fit of the distribution

Calculate the missing values in each of the following cells.

• C3

• D3

• E3

Explain why the numbers for 5, 6 and at least 7 muons have been combined into the single

category of at least 5 muons, as shown in Fig. 6.2.

In this question you must show detailed reasoning.

Carry out the test at the 5% significance level.

Explain why the physiotherapist used the same people for testing their dominant and non-

dominant grip strengths.

State any assumptions necessary in order to construct the confidence interval shown in Fig. 7.

Explain whether the confidence interval supports the physiotherapist’s belief.

The physiotherapist then finds some data which have previously been collected on grip strength using a sample of 100 adult women. A 95% confidence interval, based on this sample and calculated using a Normal distribution, for the mean difference in grip strength between the two hands (dominant minus non-dominant) is (1.94, 2.84).

For this sample, find

• the mean difference

• the standard deviation of the differences.

Explain what you would need to know about the nature of this sample if you wanted to draw conclusions about the mean difference in grip strength in the population of adult

women.

VO_2max is a measure of athletic fitness. Since VO_2max is fairly time-consuming and expensive to measure, an exercise scientist wants to predict VO_2max from data such as times for running different distances. The scientist uses these data for a random sample of 15 athletes to predict their VO_2max values, denoted by y, in suitable units. She also obtains accurate measurements of the VO_2max values, denoted by x, in the same units.

The scatter diagram in Fig. 8.1 shows the values of x and y obtained, together with the equation of the regression line of y on x and the value of r².

Use the regression line to estimate the predicted VO_2max of an athlete whose accurately

measured VO_2max is 50.

Comment on the reliability of your estimate.

The equation of the regression line of x on y is x = 0.7565 y + 10.493.

Find the coordinates of the point at which the two regression lines meet.

State what the point you found in part (iii) represents.

It is known that there is negative correlation between VO_2max and marathon times in very good runners (those whose best marathon times are under 3 hours). The exercise scientist wishes to know whether the same applies to runners who take longer to run a marathon. She selects a random sample of 20 runners whose best marathon times are between 3 hours and hours and accurately measures their VO_2max.

Fig. 8.2 is a scatter diagram of accurately measured VO_2max, v units, against best marathon time, t hours, for these runners.

Explain why the exercise scientist comes to the conclusion that a test based on Pearson’s

product moment correlation coefficient may be valid.

Summary statistics for the 20 runners are as follows.

Find the value of Pearson’s product moment correlation coefficient.

The discrete random variable X has a uniform distribution over the set of all integers between - n and n inclusive, where n is a positive integer.

Given that n is odd, determine P(X > ), giving your answer as a single fraction in terms

of n.

Determine the variance of the sum of 10 independent values of X, giving your answer in the

form an² + bn, where a and b are constants.

Use the spreadsheet output to estimate each of the following.

• P(T ⩽ 56)

• P(T > 61)

The random variable W is Normally distributed with the same mean and variance as T. Find each of the following.

• P(W ⩽ 56)

• P(W > 61)

Given that E(X) = 2, determine the values of a and b.

Determine the median value of X.

A random sample of 50 observations of X is selected.

Given that Var(X) = 0.2, determine an estimate of the probability that the mean value of the 50

observations is less than 1.9.