Edexcel A Level Further Maths Statistics 4 (S4) June 2019 question paper 6686

A factory worker has two alternative routes to work. The times taken, in minutes, on 9 randomly chosen occasions that the factory worker used route 1 are summarised by

The times taken, in minutes, on 12 randomly chosen occasions that the factory worker used route 2 are summarised by

You may assume that the times are independent and normally distributed.

Stating your hypotheses clearly, test, at the 10% level of significance, whether or not there is evidence that the variances of the times taken on route 1 and on route 2 are the same.

Stating your hypotheses clearly, test, at the 5% level of significance, whether or not there is evidence that there is a difference in the means of the times taken on the two routes. You must show your working clearly and state the critical value used.

Explain why your result in part (a) enables you to carry out the test in part (b).

An athlete is investigating the times taken, to the nearest minute, for runners to complete a 10 km road race and a 10 km cross-country race. She takes a random sample of 10 runners and records their times taken to complete each race. The results are shown below.

The athlete claims that the race times for the road race are quicker than the race times for the cross-country race.

Test, at the 1% level of significance, whether or not the results support the athlete’s claim. State your hypotheses and show your working clearly.

State a necessary assumption for the test in part (a).

When a biased six-sided die is rolled, the probability that a six will be obtained is an unknown constant p. The die is rolled n times and the number, X, of sixes obtained is recorded. The die is then rolled a further 2 n times and the number, Y, of sixes obtained is also recorded.

Show that

and

are both unbiased estimators for p.

Find the variance for each of these estimators.

State, with a reason, which of the two estimators for p is the better.

The weights of apples are normally distributed. A farmer records the weights, in grams, of a random sample of 10 apples from his orchard. Based on his sample, the 95% confidence interval for the variance of the weights of apples from his orchard is [12.31, 86.70]

Show that the sample standard deviation, s, is 5.1 grams to 1 decimal place.

Given that the mean weight of the sample of the 10 apples is 85 grams,

calculate a 99% confidence interval for the mean weight of the apples in the farmer’s orchard. Show your working clearly.

The weights of Great Dane dogs are normally distributed. A random sample of 12 Great Dane dogs is taken and their weights, to the nearest kilogram, are recorded. The results are given below. 73 68 67 69 71 72 70 77 74 72 70 69 Stating your hypotheses and showing your working clearly, test at the 5% level of significance,

whether or not the mean weight of Great Dane dogs is less than 72 kg,

whether or not the standard deviation of the weights of Great Dane dogs is 6 kg.

A factory uses a machine to fill bags with sand. The weight of sand in a bag, measured in kg, is distributed normally with a standard deviation of 2.5 and an intended mean of 38 The manager of the factory is concerned that the machine is not set correctly and it does not put enough sand into the bags. He therefore takes a random sample of 20 bags and weighs them in order to verify that the machine has been set correctly.

Write down the appropriate null and alternative hypotheses to test the manager’s concern.

Using a 5% level of significance, find the critical region for the mean weight of sand in a sample of 20 bags for this hypothesis test.

Given that the machine is filling the bags with a mean weight of sand of 37.5 kg,

find the probability of a Type II error for this test.

The manager decides to take a larger sample of size n. The manager requires the probability of a Type II error to be less than 0.2 when the machine is filling the bags with a mean weight of sand of 37.5 kg.

Calculate the minimum value of n.

A random variable, X, has a probability density function given by

where and k are positive constants. Hypotheses are set up such that

The suggested test procedure, based on a single observation on X, is to reject H₀ in favour of H₁ only if the observation is greater than 9.5

Determine the size of this test.

Given that = 2

calculate the power of this test.