Edexcel FS2 AS (Further Statistics 2 AS) 2022 June

1. Abena and Meghan are both given the same list of 10 films.

Each of them ranks the 10 films from most favourite to least favourite.
For the differences, \(d\), between their ranks for these 10 films, \(\sum d ^ { 2 } = 84\)

1. Calculate Spearman's rank correlation coefficient between Abena's ranks and Meghan's ranks. A test is carried out at the 5\% level of significance to see if there is agreement between their ranks for the films. The hypotheses for the test are $$\mathrm { H } _ { 0 } : \rho _ { \mathrm { S } } = 0 \quad \mathrm { H } _ { 1 } : \rho _ { \mathrm { S } } > 0$$
2. 1. Find the critical region for the test.
   2. State the conclusion of the test. An 11th film is added to the list. Abena and Meghan both agree that this film is their least favourite. A new test is carried out at the \(5 \%\) level of significance using the same hypotheses.
3. Determine the conclusion of this test. You should state the test statistic and the critical value used.

1. The graph shows the probability density function \(\mathrm { f } ( x )\) of the continuous random variable \(X\)
   \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-04_951_1365_322_331}
   1. Find \(\mathrm { P } ( X < 4 )\)
   2. Specify the cumulative distribution function of \(X\) for \(7 \leqslant x \leqslant 11\)

1. Gabriela is investigating a particular type of fish, called bream. She wants to create a model to predict the weight, \(w\) grams, of bream based on their length, \(x \mathrm {~cm}\).

For a sample of 27 bream, some summary statistics are given below. $$\begin{gathered} \bar { x } = 31.07 \quad \bar { w } = 628.59 \quad \sum w ^ { 2 } = 11386134
\mathrm {~S} _ { x w } = 13082.3 \quad \mathrm {~S} _ { x x } = 260.8 \end{gathered}$$

1. Find the value of the product moment correlation coefficient between \(x\) and \(w\)
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 \(w\) on \(x\) in the form \(w = a + b x\) A residual plot for these data is shown below.
   \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-06_931_1790_1107_139} One of the bream in the sample has a length of 32 cm .
4. Find its weight.
5. With reference to the residual plot, comment on the model for bream with lengths above 33 cm .

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

$$f ( x ) = \left\{ \begin{array} { c c } 0.8 - 6.4 x ^ { - 3 } & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$ The median of \(X\) is \(m\)

1. Show that \(m ^ { 3 } - 3.625 m ^ { 2 } + 4 = 0\)
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\)
   2. Explain why the mode of \(X\) is 4 Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 10.5\) to 3 significant figures,
3. find \(\operatorname { Var } ( X )\), showing your working clearly.

1. The random variable \(X\) has the continuous uniform distribution over the interval [0.5, 2.5]

Talia selects a number, \(T\), at random from the distribution of \(X\)

1. Find \(\mathrm { P } ( T < 1 )\) Malik takes Talia's number, \(T\), and calculates his number, \(M\), where \(M = \frac { 1 } { T ^ { 2 } }\)
2. Find the probability that both \(T\) and \(M\) are less than 2.25 Raja and Greta play a game many times.
   Each time they play they use a number, \(R\), randomly selected from the distribution of \(X\)
   Raja's score is \(R\)
   Greta's score is \(G\), where \(G = \frac { 2 } { R ^ { 2 } }\)
3. Determine, giving a reason, who you would expect to have the higher total score.