4 In this question the product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
- 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. - On the diagram in the Answer Booklet, draw five points such that \(r _ { s } = 1\) and \(r \neq 1\).
- 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 \(r _ { s }\). - A random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.6 )\). Find
- \(\mathrm { P } ( X \leqslant 14 )\),
- \(\mathrm { P } ( X = 14 )\),
- \(\quad \operatorname { Var } ( X )\).
- 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 \(y = 8\).
- 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.
- Find the number of possible arrangements of the 7 letters.
- 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.
- Find the probability that the letters form the word ABBA .