Edexcel FS2 AS (Further Statistics 2 AS) 2023 June

1. Every applicant for a job at Donala is given three different tasks, \(P , Q\) and \(R\).

For each task the applicant is awarded a score.
The scores awarded to 9 of the applicants, for the tasks \(P\) and \(Q\), are given below.

1. Calculate Spearman's rank correlation coefficient for the scores awarded for the tasks \(P\) and \(Q\).
2. Test, at the \(1 \%\) level of significance, whether or not there is evidence for a positive correlation between the ranks of scores for tasks \(P\) and \(Q\). You should state your hypotheses and critical value clearly. The Spearman's rank correlation coefficient for \(P\) and \(R\) is 0.290 and for \(Q\) and \(R\) is 0.795 The manager of Donala wishes to reduce the number of tasks given to job applicants from three to two.
3. Giving a reason for your answer, state which 2 tasks you would recommend the manager uses.

1. A continuous random variable \(X\) has probability density function

$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 16 } \left( 9 - x ^ { 2 } \right) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$

1. Find the cumulative distribution function of \(X\)
2. Calculate \(\mathrm { P } ( X > 1.8 )\)
3. Use calculus to find \(\mathrm { E } \left( \frac { 3 } { X } + 2 \right)\)
4. Show that the mode of \(X\) is \(\sqrt { 3 }\)

1. Pat is investigating the relationship between the height of professional tennis players and the speed of their serve. Data from 9 randomly selected professional male tennis players were collected. The variables recorded were the height of each player, \(h\) metres, and the maximum speed of their serve, \(v \mathrm {~km} / \mathrm { h }\).

Pat summarised these data as follows $$\sum h = 17.63 \quad \sum v = 2174.9 \quad \sum v ^ { 2 } = 526407.8 \quad S _ { h h } = 0.0487 \quad S _ { h v } = 5.1376$$

1. Calculate the product moment correlation coefficient between \(h\) and \(v\)
2. Explain whether the answer to part (a) is consistent with a linear model for these data.
3. Find the equation of the regression line of \(v\) on \(h\) in the form \(v = a + b h\) where \(a\) and \(b\) are to be given to one decimal place. Pat calculated the sum of the residuals for the 9 tennis players as 1.04
4. Without doing a calculation, explain how you know Pat has made a mistake. Pat made one mistake in the calculation. For the tennis player of height 1.96 m Pat misread the residual as 2.27
5. Find the maximum speed of serve, in km/h, for the tennis player of height 1.96 m

1. The random variable \(X\) has a continuous uniform distribution over the interval \([ - 3 , k ]\) Given that \(\mathrm { P } ( - 4 < X < 2 ) = \frac { 1 } { 3 }\)
   1. find the value of \(k\)
   A computer generates a random number, \(Y\), where
   • \(\quad Y\) has a continuous uniform distribution over the interval \([ a , b ]\)
   • \(\mathrm { E } ( Y ) = 6\)
   • \(\operatorname { Var } ( Y ) = 192\)
   The computer generates 5 random numbers.
2. Calculate the probability that at least 2 of the 5 numbers generated are greater than 7.5