Resit questions
Answer all 21 questions. There are 108 marks available.
Show all necessary working. Answers without necessary working will be penalised.
A file of mathematical formulae and tables, including statistical tables, is provided. Apart from the Normal distribution, these statistical tables are the same as those used in the course. If you need to use a table of the Normal distribution, you may use either the table in the statistical tables provided with the coursework or the version of the Normal distribution used in the course. Statistical table values provided directly from a computer or calculator may not be the same as the values in the tables described here, so you will not be able to demonstrate all your necessary working, making your answer liable to a penalty.
Answers should be hand written. Typed answers will not be marked.
Question 1
For the events J and K:
P(J∪K) = 0.5, P(J ′∩K) = 0.2, P(J∩K ′) = 0.25.
a Draw a Venn diagram to represent the events J and K and the sample space S.
Find:
b P(J)
c P(K)
d P(J|K)
e Determine whether or not J and K are independent.
(7 marks)
Question 2
The scores of eighteen people who take a quiz are:
34 26 30 35 22 36
34 38 41 35 28 34
30 29 39 22 37 36
Calculate the mean, variance and standard deviation of the scores.
(4 marks)
Question 3
Patsy collects some data to find out if there is any relationship between the numbers of car accidents and computer ownership. She calculates the product moment correlation coefficient between the two variables. There is a strong positive correlation. She says that as car accidents increase so does computer ownership. Comment, giving your reasons, on whether this statement is justified.
(2 marks)
Question 4
A discrete random variable has a cumulative distribution function F(x) given in the table.
X 0 1 2 3 4 5 6
F(x) 0 0.1 0.2 0.45 0.5 0.9 1.0
a Draw up a table to show the probability distribution X.
b Write down P(X < 5).
c Find P(2 ≤ X < 5).
(4 marks)
Question 5
The weights of batches of corrugated board dispatched from a corrugating plant are known to be normally distributed with mean 32.5 kg and standard deviation 2.2 kg
a Find the percentage of batches that weigh less than 30 kg.
b Find the percentage of batches that weigh between 31.6 kg and 34.8 kg.
(5 marks)
Question 6
The continuous random variable X has probability density function given by:
f(x) = kx3 1 ≤ x ≤ 6
0 otherwise
Find the value of k.
(3 marks)
Question 7
A fair coin is tossed 80 times. Use a suitable approximation to estimate the probability of obtaining more than 50 heads.
(3 marks)
Question 8
A marina hires out boats on a daily basis. The marina owns 22 boats and the mean number of boats hired each day is 16 (boats are hired out for the whole day). Using a normal approximation to the Poisson distribution, find for a period of 120 days:
a the expected value of the number of days on which exactly 12 boats are hired out
b the expected value of the number of days on which customers will have to be turned away.
(6 marks)
Question 9
A random sample M1, M2, M3,..M8 is taken from a population with unknown mean µ. For each of the following, state whether or not it is a statistic, giving your reasons.
a (M3 + M8)/2
b where n is the sample size and summation is over all the observations in the sample.
(4 marks)
Question 10
The success rate of the standard treatment for patients suffering from a particular disease is claimed to be 68%.
a In a sample of n patients, X is the number of patients for whom the treatment is successful. Write down a suitable distribution for X. Give reasons for your choice.
A random sample of 10 patients receives the standard treatment and the treatment is successful in only 3 cases. There are concerns that the standard treatment is not as effective as it is claimed to be.
b Test the claim about the effectiveness of the standard treatment at the 5% level of significance.
(8 marks)
Question 11
Over a number of years the mean number of hurricanes experienced in a certain area during the month of August is four. A scientist suggests that, due to global warming, the number of hurricanes may have increased, and proposes to do a hypothesis test based on the number of hurricanes in August this year.
a Suggest suitable hypotheses for this test.
b Find what number of hurricanes is needed for the null hypothesis to be rejected at the 1% level of significance.
c What conclusion does the scientist reach if the actual number of hurricanes this year is 10?
(6 marks)
Question 12
X1 and X2 are independent normal random variables. X1 ~ N(60,25) and X2 ~ N(50,16). Find the distribution of T where:
a T = 3X1
b T = 3X1 + 7X2
(4 marks)
Question 13
Explain:
a what a sampling frame is.
b what effect the size of the population has on the size of the sampling frame.
c what effect the variability of the population has on the size of the sampling frame.
(6 marks)
Question 14
A biased six-sided die has a probability p of landing showing a six. Every day, for a period of 25 days, the die is rolled 10 times and the number of sixes X is recorded, giving rise to a sample X1, X2,..X25
a Write down E(X) in terms of p.
b Show that the sample mean is a biased estimator of p and find the bias.
c Suggest a suitable unbiased estimator of p.
(4 marks)
Question 15
A machine operator checks a random sample of 20 bottles from a filling line and measures in cm3 the volume of liquid x in each bottle. The 20 values can be summarised as ∑ x = 1300 and ∑ x2 = 84685.
a Use this sample to find unbiased estimates of μ and 𝜎2, the mean and variance of the volumes.
The machine operator takes a further sample of volumes from 16 bottles and finds ∑x = 1060.
b Combine the two samples to obtain a revised estimate of μ.
The supervisor on a different filling line knows from experience that the standard deviationof the volume filled in each bottle on his line is 3 cm3, and wants an estimate of the mean volume filled on his line that has a standard error of less than 0.5 cm3.
c What size sample will be needed to achieve this?
(11 marks)
Question 16
How many times must a fair die be rolled for there to be probability of at least 99% that the mean score is between 3.4 and 3.6?
(5 marks)
Question 17
The breaking stresses of rubber bands are normally distributed. A company uses rubber bands with a mean breaking stress of 46.5 N. A new supplier claims that it can supply bands that are stronger and provides a sample of 100 bands for the company to test. The company checks the breaking stress X for each of these 100 bands and the results are summarised as: ∑x = 4715, ∑x2 = 222910.
a Find an approximate 95% confidence interval for the mean breaking stress of these new rubber bands.
b Test at the 5% level whether or not there is evidence that the new rubber bands are stronger.
(7 marks)
Question 18
Find the value of χ210 that has a probability of 0.10 of being exceeded.
(2 marks)
Question 19
The numbers of trains arriving on time and late were observed at three different London stations. The results were:
Station On time Late
A 26 14
B 30 10
C 44 26
Using a χ2 statistic and testing at the 5% significance level, check whether there is evidence that lateness varies between stations.
(6 marks)
Question 20
The ages X (in years) and weights Y (in pounds) of 11 members of a football team were recorded and the following statistics were calculated:
∑x = 168
∑y= 1275
∑xy = 20704
∑x2 = 2585
∑y2 = 320019
a Calculate the product moment correlation coefficient for these data.
b Test at the 5% level the proposition that age and weight are positively correlated. State your assumptions as well as your conclusions.
(7 marks)
Question 21
Describe the conditions under which you would measure association using a rank correlation coefficient rather than a product moment correlation coefficient.
(4 marks)
