OCR Further Statistics AS (Further Statistics AS) 2018 June

1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the Xth weekday of September.

1. State an assumption needed for \(X\) to be well modelled by a geometric distribution.
2. Find \(\mathrm { P } ( X = 11 )\).
3. Find \(\mathrm { P } ( X \leqslant 8 )\).
4. Find \(\operatorname { Var } ( X )\).
5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.

2 The probability distribution for the discrete random variable \(W\) is given in the table.

1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
2. Find \(\operatorname { Var } ( 3 W + 6 )\).

3 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).

1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4.

4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$

1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
3. Assume now that the assumption in part (ii) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.

5

1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
   Find the probability that the team contains no more than 3 girls.
2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).

6 In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(\mathrm { E } ( T ) = 5.625\) and the standard deviation of \(T\) is 1.875 . Find the values of the parameters of the distribution.

7 An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.

1. State which, if either, of \(m\) and \(c\) is an independent variable.
2. Calculate the equation of the least squares regression line of \(c\) on \(m\).
3. State what effect, if any, there would be on your answer to part (ii) if the masses of the fish had been recorded in kilograms rather than pounds. ( \(1 \mathrm {~kg} \approx 2.2\) pounds.)
4. The data is illustrated in the scatter diagram. Explain what is meant by 'least squares', illustrating your answer using the copy of this diagram in the Printed Answer Booklet.
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8 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\). Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}