OCR Further Statistics AS (Further Statistics AS) Specimen

1 Two music critics, \(P\) and \(Q\), give scores to seven concerts as follows.

1. Calculate Spearman's rank correlation coefficient, \(r _ { s }\), for these scores.
2. Without carrying out a hypothesis test, state what your answer tells you about the views of the two critics.

2 The probability distribution of a discrete random variable \(W\) is given in the table. Given that \(\mathrm { E } ( W ) = 1.61\), find the value of \(\operatorname { Var } ( 3 W + 2 )\).

3 Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below. Test at the 5\% significance level whether Carl's belief is justified.

4

1. Four men and four women stand in a random order in a straight line. Determine the probability that no one is standing next to a person of the same gender.
2. \(x\) men, including Mr Adam, and \(x\) women, including Mrs Adam, are arranged at random in a straight line. Show that the probability that Mr Adam is standing next to Mrs Adam is \(\frac { 1 } { X }\).

1. The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.6 )\).
   (a) Find \(\mathrm { P } ( X \geq 8 )\).
   (b) Find the value of \(\mathrm { E } ( X )\).
   (c) Find the value of \(\operatorname { Var } ( X )\).
2. The random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\mathrm { P } ( Y < 4 ) = 0.986\) correct to 3 significant figures. Use an algebraic method to find the value of \(p\). Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
3. State these two assumptions.
4. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\operatorname { Po } ( 0.8 )\).
5. (a) Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house.
   (b) Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house.
6. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house.
7. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition.

7 The discrete random variable \(X\) is equally likely to take values 0,1 and 2 .
\(3 N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0 , X = 1\) and \(X = 2\) are given in the following table. The test statistic for a chi-squared goodness of fit test for the data is 0.3 . Find the value of \(N\).

8 The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.

1. Find the equation of the regression line of \(y\) on \(x\). You are given that the product moment correlation coefficient is 0.959 .
2. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.
3. Assume that the data is a random sample of all years.
   (a) Carry out the test at the \(10 \%\) significance level.
   (b) Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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