6 The masses, \(x\) grams, of 800 apples are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-4_592_1363_957_351}
- On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\).
- Find an estimate of the median mass of the apples.
- Two judges rank \(n\) competitors, where \(n\) is an even number. Judge 2 reverses each consecutive pair of ranks given by Judge 1, as shown.
| Competitor | \(C _ { 1 }\) | \(C _ { 2 }\) | \(C _ { 3 }\) | \(C _ { 4 }\) | \(C _ { 5 }\) | \(C _ { 6 }\) | \(\ldots \ldots\) | \(C _ { n - 1 }\) | \(C _ { n }\) |
| Judge 1 rank | 1 | 2 | 3 | 4 | 5 | 6 | \(\ldots \ldots\) | \(n - 1\) | \(n\) |
| Judge 2 rank | 2 | 1 | 4 | 3 | 6 | 5 | \(\ldots \ldots\) | \(n\) | \(n - 1\) |
- An experiment produced some data from a bivariate distribution. The product moment correlation coefficient is denoted by \(r\), and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
(a) Explain whether the statement
$$r = 1 \Rightarrow r _ { s } = 1$$
is true or false.
(b) Use a diagram to explain whether the statement
$$r \neq 1 \Rightarrow r _ { s } \neq 1$$
is true or false.
8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 . - Find the probability that
(a) the first time she succeeds is on her 5th attempt,
(b) the first time she succeeds is after her 5th attempt,
(c) the second time she succeeds is before her 4th attempt.
Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first. - Find the probability that the first person to hit the target is Sandra, on her
(a) 2nd attempt,
(b) 10th attempt.
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