Edexcel S1 (Statistics 1)

1. (a) Explain briefly what is meant by a random variable.
   (b) Write down a quantity which could be modelled as
   1. a discrete random variable,
   2. a continuous random variable.
   3. The discrete random variable \(X\) has the probability function given by the following table:

   \(x\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( X = x )\)	0.09	0.12	0.22	0.16	\(p\)	\(2 p\)	0.2

   (a) Show that \(p = 0.07\)
   (b) Find the value of \(\mathrm { E } ( X + 2 )\).
   (c) Find the value of \(\operatorname { Var } ( 3 X - 1 )\).

3. Twenty pairs of observations are made of two variables \(x\) and \(y\), which are believed to be related. It is found that $$\sum x = 200 , \quad \sum y = 174 , \quad \sum x ^ { 2 } = 6201 , \quad \sum y ^ { 2 } = 5102 , \quad \sum x y = 5200 .$$ Find

1. the product-moment correlation coefficient between \(x\) and \(y\),
2. the equation of the regression line of \(y\) on \(x\). Given that \(p = x + 30\) and \(q = y + 50\),
3. find the equation of the regression line of \(q\) on \(p\), in the form \(q = m p + c\).
4. Estimate the value of \(q\) when \(p = 46\), stating any assumptions you make.

4. The heights of the students at a university are assumed to follow a normal distribution. \(1 \%\) of the students are over 200 cm tall and 76\% are between 165 cm and 200 cm tall. Find

1. the mean and the variance of the distribution,
2. the percentage of the students who are under 158 cm tall.
3. Comment briefly on the suitability of a normal distribution to model such a population. \section*{STATISTICS 1 (A) TEST PAPER 3 Page 2}

1. In a survey of natural habitats, the numbers of trees in sixty equal areas of land were recorded, as follows:

1. Construct a stem-and-leaf diagram to illustrate this data, using the groupings 5-9, 10-14, 15-19, 20-24, etc.
2. Find the three quartiles for the distribution.
3. On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers.

6. Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac { 1 } { 3 }\).

1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r ( r - 1 ) = 420$$
2. Hence or otherwise find the value of \(r\).
3. Find the probability that, when three cards are drawn at random from the pack,
   1. at least two are red,
   2. the first one is red given that at least two are red.