AQA A Level Further Maths Paper 3S June 2019 question paper 7367/3S

The discrete random variable X has Var (X) 5

Circle your answer.

17 20 77 80

Amy takes a sample of size 50 from a normal distribution with mean and variance 16

She conducts a hypothesis test with hypotheses:

She rejects the null hypothesis if her sample has a mean greater than 53

The actual population mean is 53.5

Find the probability that Amy makes a Type II error.

Circle your answer.

0.4% 3.9% 18.9% 45.0%

Alan’s journey time to work can be modelled by a normal distribution with standard deviation 6 minutes.

Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes.

Construct a 95% confidence interval for Alan’s mean journey time to work, giving your values to one decimal place.

Alan claims that his mean journey time to work is 30 minutes.

State, with a reason, whether or not the confidence interval found in part (a) supports Alan’s claim.

Suppose that the standard deviation is not known but a sample standard deviation is found from Alan’s sample and calculated to be 6

Explain how the working in part (a) would change.

A random variable X has a rectangular distribution.

The mean of X is 3 and the variance of X is 3

Determine the probability density function of X.

Fully justify your answer.

A 6 metre clothes line is connected between the point P on one building and the point Q on a second building.

Roy is concerned the clothes line may break. He uses the random variable X to model the distance in metres from P where the clothes line breaks.

State a criticism of Roy’s model.

On the axes below, sketch the probability density function for an alternative model for the clothes line.

The median claim is £200

Find P(X 2000), giving your answer to three significant figures.

The insurance company finds that the maximum possible claim is £2000 and they decide to refine their probability density function.

Suggest how this could be done.

During August, 102 candidates took their driving test at centre A and 60 passed.

During the same month, 110 candidates took their driving test at centre B and 80 passed.

Test whether the driving test result is independent of the driving test centre using the 5% level of significance.

Rebecca claims that if the result of the test in part (a) is to reject the null hypothesis then it is easier to pass a driving test at centre B than centre A.

State, with a reason, whether or not you agree with Rebecca’s claim.

The number of telephone calls received by an office can be modelled by a Poisson distribution with mean 3 calls per 10 minutes.

Find the probability that:

the office receives exactly 2 calls in 10 minutes;

the office receives more than 30 calls in an hour.

The office manager splits an hour into 6 periods of 10 minutes and records the number of telephone calls received in each of the 10 minute periods.

Find the probability that the office receives exactly 2 calls in a 10 minute period exactly twice within an hour.

The office has just received a call.

Find the probability that the next call is received more than 10 minutes later.

Mahah arrives at the office 5 minutes after the last call was received.

State the probability that the next call received by the office is received more than 10 minutes later.

Explain your answer.