4 In this question the product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).

1. The scatter diagram in Fig. 1 shows the results of an experiment involving some bivariate data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_597_595_434_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Write down the value of \(r _ { s }\) for these data.
2. On the diagram in the Answer Booklet, draw five points such that \(r _ { s } = 1\) and \(r \neq 1\).
3. The scatter diagram in Fig. 2 shows the results of another experiment involving 5 items of bivariate data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_604_608_1484_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Calculate the value of \(r _ { s }\).
4. A random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.6 )\). Find
   (a) \(\mathrm { P } ( X \leqslant 14 )\),
   (b) \(\mathrm { P } ( X = 14 )\),
   (c) \(\quad \operatorname { Var } ( X )\).
5. A random variable \(Y\) has the distribution \(\mathrm { B } ( 24,0.3 )\). Write down an expression for \(\mathrm { P } ( Y = y )\) and evaluate this probability in the case where \(y = 8\).
6. A random variable \(Z\) has the distribution \(\mathrm { B } ( 2,0.2 )\). Find the probability that two randomly chosen values of \(Z\) are equal.
   (a) Find the number of ways in which 12 people can be divided into three groups containing 5 people, 4 people and 3 people, without regard to order.
   (b) The diagram shows 7 cards, each with a letter on it. $$\mathrm { A } \mathrm {~A} \mathrm {~A} \mathrm {~B} \text { } \mathrm { B } \text { } \mathrm { R } \text { } \mathrm { R }$$ The 7 cards are arranged in a random order in a straight line.
7. Find the number of possible arrangements of the 7 letters.
8. Find the probability that the 7 letters form the name BARBARA. The 7 cards are shuffled. Now 4 of the 7 cards are chosen at random and arranged in a random order in a straight line.
9. Find the probability that the letters form the word ABBA .