Edexcel FS2 AS (Further Statistics 2 AS) 2020 June

1. An estate agent in Tornep believes that houses further from the railway station are more expensive than those that are closer. She took a random sample of 22 three-bedroom houses in Tornep and calculated the product moment correlation coefficient between the house price and the distance from the station to be 0.3892

Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the estate agent's belief. State the critical region used in your test.

1. Mary, Jahil and Dawn are judging the cakes in a village show. They have 5 features to consider and each feature is awarded up to 5 points. The total score the judges gave each cake are given in the table below.

1. Calculate Spearman's rank correlation coefficient between Mary's scores and Jahil’s scores.
2. Calculate Spearman's rank correlation coefficient between Jahil's scores and Dawn's scores. The judges discussed their interpretation of the points system and agreed that the first prize should go to cake \(C\).
3. Explain how different interpretations of the points system could give rise to the results in part (a) and part (b).

1. The continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 4
p x - k \sqrt { x } & 4 \leqslant x \leqslant 9
1 & x > 9 \end{array} \right.$$ where \(p\) and \(k\) are constants.

1. Find the value of \(p\) and the value of \(k\). Given that \(\mathrm { E } ( X ) = \frac { 119 } { 18 }\)
2. show that \(\operatorname { Var } ( X ) = 2.05\) to 3 significant figures.
3. Write down the mode of \(X\).
4. Find the exact value of the constant \(a\) such that \(\mathrm { P } ( X \leqslant a ) = \frac { 7 } { 27 }\)

1. Some students are investigating the strength of wire by suspending a weight at the end of the wire. They measure the diameter of the wire, \(d \mathrm {~mm}\), and the weight, \(w\) grams, when the wire fails. Their results are given in the following table.

The first 14 points are plotted on the axes on page 13.

1. On the axes on page 13, complete the scatter diagram for these data.
2. Use your calculator to write down the equation of the regression line of \(w\) on \(d\).
3. With reference to the scatter diagram, comment on the appropriateness of using this linear regression model to make predictions for \(w\) for different values of \(d\) between 0.5 and 5.4 The product moment correlation coefficient for these data is \(r = 0.987\) (to 3 significant figures).
4. Calculate the residual sum of squares (RSS) for this model. Robert, one of the students, suggests that the model could be improved and intends to find the equation of the line of regression of \(w\) on \(u\), where \(u = d ^ { 2 }\)
   He finds the following statistics $$\mathrm { S } _ { w u } = 5721.625 \quad \mathrm {~S} _ { u u } = 1482.619 \quad \sum u = 157.57$$
5. By considering the physical nature of the problem, give a reason to support Robert's suggestion.
6. Find the equation of the regression line of \(w\) on \(u\).
7. Find the residual sum of squares (RSS) for Robert's model.
8. State, giving a reason based on these calculations, which of these models better describes these data.
   1. Hence estimate the weight at which a piece of wire with diameter 3 mm will fail. \begin{figure}[h]
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      \end{figure}