- \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
- if \(X\) and \(Y\) are independent then \(\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )\).
\section*{Non-parametric tests}
Goodness-of-fit test and contingency tables: \(\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } } \sim \chi _ { v } ^ { 2 }\)
Approximate distributions for large samples
Wilcoxon Signed Rank test: \(T \sim \mathrm {~N} \left( \frac { 1 } { 4 } n ( n + 1 ) , \frac { 1 } { 24 } n ( n + 1 ) ( 2 n + 1 ) \right)\)
Wilcoxon Rank Sum test (samples of sizes \(m\) and \(n\), with \(m \leq n\) ):
$$W \sim \mathrm {~N} \left( \frac { 1 } { 2 } m ( m + n + 1 ) , \frac { 1 } { 12 } m n ( m + n + 1 ) \right)$$
\section*{Percentage points of the normal distribution}
If \(Z\) has a normal distribution with mean 0 and variance 1 then, for each value of \(p\), the table gives the value of \(z\) such that \(P ( Z \leq z ) = p\).
Test whether there is evidence, at the \(1 \%\) significance level, that the judges agree with each another.
The marks given by the two judges to the 7 randomly chosen contestants were as follows, where \(x\) is an integer.
| Contestant | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) |
| Judge 1 | 64 | 65 | 67 | 78 | 79 | 80 | 86 |
| Judge 2 | 61 | 63 | 78 | 80 | 81 | 90 | \(x\) |
(b) Use the value \(r _ { s } = \frac { 27 } { 28 }\) to determine the range of possible values of \(x\).
(c) Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context.
Total: \(\_\_\_\_\) / 9 marks
\section*{Question 7}
A circle, centre \(O\), has radius \(x \mathrm {~cm}\), where \(x\) is an observation from the random variable \(X\) which has a uniform distribution on \([ 0 , \pi ]\)
(a) Find the probability that the area of the circle is greater than \(10 \mathrm {~cm} ^ { 2 }\)
(b) State, giving a reason, whether the median area of the circle is greater or less than \(10 \mathrm {~cm} ^ { 2 }\)
The triangle \(O A B\) is drawn inside the circle with \(O A\) and \(O B\) as radii of length \(x \mathrm {~cm}\) and angle \(A O B\) is \(x\) radians.
(c) Use algebraic integration to find the expected value of the area of triangle \(O A B\). Give your answer as an exact value. Detailed working is required and calculator integration is not allowed.
[0pt]
[6]
Total: \(\_\_\_\_\) / 10 marks
End of Paper