Edexcel FS2 AS (Further Statistics 2 AS) 2024 June

1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < - 1
\frac { 1 } { 5 } ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 0
1 - \frac { 1 } { 20 } ( 4 - x ) ^ { 2 } & 0 < x \leqslant 4
1 & x > 4 \end{array} \right.$$

1. Find the probability density function, \(\mathrm { f } ( x )\)
2. 1. Sketch \(\mathrm { f } ( x )\)
   2. Hence describe the skewness of the distribution.
3. Find, to 3 significant figures, the value of \(c\) such that $$\mathrm { P } ( 1 < X < c ) = \mathrm { P } ( c < X < 2 )$$

1. A random sample of size \(n = 8\) of paired data is taken from a population. The data are plotted below.
   \includegraphics[max width=\textwidth, alt={}, center]{ba41c616-0805-4466-81b8-b985b0bdd94b-06_572_983_335_541}

Test, at the \(1 \%\) level of significance, whether or not there is evidence of a negative rank correlation between the two variables. You should state your hypotheses and critical value and show your working clearly.

1. The continuous random variable \(Y\) has probability density function

$$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise } \end{array} \right.$$

1. Show that the mode of \(Y\) is 1 , justifying your reasoning. Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
2. determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer. Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
3. find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)

1. The continuous random variable \(X\) is uniformly distributed over the interval [2, 7]
   1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find \(\mathrm { P } ( 1 < X < 4 )\)
   3. Find \(\mathrm { P } \left( 2 X ^ { 2 } - 15 X + 27 > 0 \right)\)
   4. Find \(\mathrm { E } \left( \frac { 3 } { X ^ { 2 } } \right)\)

1. A random sample of 24 adults is taken. The height, \(h\) metres, and the arm span, \(s\) metres, for each adult are recorded.

These data are summarised below. $$\mathrm { S } _ { h h } = 0.377 \quad \mathrm {~S} _ { s h } = 0.352 \quad \bar { s } = 1.70 \quad \bar { h } = 1.68$$ The least squares regression line of \(h\) on \(s\) is $$h = a + 0.919 s$$ where \(a\) is a constant.

1. Calculate the product moment correlation coefficient. A doctor uses the least squares regression line of \(h\) on \(s\) as a model to predict a person's height based on their arm span.
2. Use the model to predict the height of an adult with arm span 1.79 metres. Ewan has an arm span of 1.70 metres and a height of 1.75 metres. His information is added to the sample as the 25th adult.
3. Explain how the gradient of the regression line for the sample of 25 adults compares with the gradient of the regression line for the original sample of 24 adults.
   Give a reason for your answer.