OCR S1 (Statistics 1) 2005 June

1

1. Calculate the value of Spearman's rank correlation coefficient between the two sets of rankings, \(A\) and \(B\), shown in Table 1. \begin{table}[h]

   \(A\)	1	2	3	4	5
   \(B\)	4	1	3	2	5

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. The value of Spearman's rank correlation coefficient between the set of rankings \(B\) and a third set of rankings, \(C\), is known to be - 1 . Copy and complete Table 2 showing the set of rankings \(C\). \begin{table}[h]

   \(B\)	4	1	3	2	5
   \(C\)

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}

2 The probability that a certain sample of radioactive material emits an alpha-particle in one unit of time is 0.14 . In one unit of time no more than one alpha-particle can be emitted. The number of units of time up to and including the first in which an alpha-particle is emitted is denoted by \(T\).

1. Find the value of
   (a) \(\mathrm { P } ( T = 5 )\),
   (b) \(\mathrm { P } ( T < 8 )\).
2. State the value of \(\mathrm { E } ( T )\).

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p .16\) shoppers are selected at random.

1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
   (a) at least 8,
   (b) between 4 and 9 inclusive.
2. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 .

4 The table shows the latitude, \(x\) (in degrees correct to 3 significant figures), and the average rainfall \(y\) (in cm correct to 3 significant figures) of five European cities. $$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$

1. Calculate the product moment correlation coefficient.
2. The values of \(y\) in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54 . State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
3. It is required to estimate the annual rainfall at Bergen, where \(x = 60.4\). Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate.

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve.
\includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-4_1344_1335_386_425} Use the curve to estimate

1. the interquartile range of the marks,
2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
5. What does this information suggest about the estimate of the interquartile range found in part (i)?

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen.
\includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-5_863_986_559_612}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).