Edexcel FS2 AS (Further Statistics 2 AS) 2019 June

1. Bara is investigating whether or not the two judges of a skating competition are in agreement. The two judges gave a score to each of the 8 skaters in the competition as shown in the table below.

Bara decided to calculate Spearman's rank correlation coefficient for these data.

1. Calculate Spearman's rank correlation coefficient between the ranks of the two judges.
2. Test, at the \(1 \%\) level of significance, whether or not the two judges are in agreement. Judge 1 accidentally swapped the scores for skaters \(D\) and \(E\). The score for skater \(D\) should be 63 and the score for skater \(E\) should be 62
3. Without carrying out any further calculations, explain how Spearman's rank correlation coefficient will change. Give a reason for your answer.

1. Lloyd regularly takes a break from work to go to the local cafe. The amount of time Lloyd waits to be served, in minutes, is modelled by the continuous random variable \(T\), having probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { t } { 120 } & 4 \leqslant t \leqslant 16
0 & \text { otherwise } \end{array} \right.$$

1. Show that the cumulative distribution function is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c r } 0 & t < 4
   \frac { t ^ { 2 } } { 240 } - c & 4 \leqslant t \leqslant 16
   1 & t > 16 \end{array} \right.$$ where the value of \(c\) is to be found.
2. Find the exact probability that the amount of time Lloyd waits to be served is between 5 and 10 minutes.
3. Find the median of \(T\).
4. Find the value of \(k\) such that $$\mathrm { P } ( T < k ) = \frac { 2 } { 3 } \mathrm { P } ( T > k )$$ giving your answer to 3 significant figures.

1. Two students, Jim and Dora, collected data on the mean annual rainfall, \(w \mathrm {~cm}\), and the annual yield of leeks, \(l\) tonnes per hectare, for 10 years.

Jim summarised the data as follows $$\mathrm { S } _ { w l } = 42.786 \quad \mathrm {~S} _ { w w } = 9936.9 \quad \sum l ^ { 2 } = 26.2326 \quad \sum l = 16.06$$

1. Find the product moment correlation coefficient between \(l\) and \(w\) Dora decided to code the data first using \(s = w - 6\) and \(t = l - 20\)
2. Write down the value of the product moment correlation coefficient between \(s\) and \(t\). Give a justification for your answer. Dora calculates the equation of the regression line of \(t\) on \(s\) to be \(t = 0.00431 s - 18.87\)
3. Find the equation of the regression line of \(l\) on \(w\) in the form \(l = a + b w\), giving the values of \(a\) and \(b\) to 3 significant figures.
4. Use your equation to estimate the yield of leeks when \(w\) is 100 cm .
5. Calculate the residual sum of squares. The graph shows the residual for each value of \(l\)
   \includegraphics[max width=\textwidth, alt={}, center]{7e46e14a-0f5a-4d02-8f00-a92bc4def6d7-08_716_1594_1594_239}
6. 1. State whether this graph suggests that the use of a linear regression model is suitable for these data. Give a reason for your answer.
   2. Other than collecting more data, suggest how to improve the fit of the model in part (c) to the data.

1. The random variable \(X\) has a continuous uniform distribution over the interval [5,a], where \(a\) is a constant.
   Given that \(\operatorname { Var } ( X ) = \frac { 27 } { 4 }\)
   1. show that \(a = 14\)
   The continuous random variable \(Y\) has probability density function $$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 20 } ( 2 y - 3 ) & 2 \leqslant y \leqslant 6
   0 & \text { otherwise } \end{array} \right.$$ The random variable \(T = 3 \left( X ^ { 2 } + X \right) + 2 Y\)
2. Show that \(\mathrm { E } ( T ) = \frac { 9857 } { 30 }\)