3 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.
\begin{enumerate}[label=(\alph*)]
\item For all sixteen candidates, the value of the product moment correlation coefficient $r$ for the marks on the two papers was 0.701 correct to 3 significant figures.

Test whether there is evidence, at the $5 \%$ significance level, of association between the marks on the two papers.
\item A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by\\
$n = 10 , \Sigma x = 750 , \Sigma y = 690 , \Sigma x ^ { 2 } = 57690 , \Sigma y ^ { 2 } = 49676 , \Sigma x y = 50829$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of $r$ for these 10 candidates.
\item What do the two values of $r$, in parts (a) and (b)(i), tell you about the scores of the sixteen candidates?

A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by $p$.\\
(a) Sasha selects 10 beads at random, with replacement.

Write down an expression, in terms of $p$, for the variance of the number of green beads Sasha selects.

Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the $X$ th bead that Freda selects.\\
(b) Assume that $p = 0.3$.

Find
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { P } ( X \geqslant 5 )$,
\item $\operatorname { Var } ( X )$.
\end{enumerate}
\end{enumerate}\item In fact, on the basis of a large number of observations of $X$, it is found that $\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )$.

Estimate the value of $p$.
\end{enumerate}

\hfill \mbox{\textit{OCR FS1 AS 2021 Q3 [5]}}