Edexcel S1 (Statistics 1)

1. Explain briefly what is meant by a random variable. [2 marks]
2. Write down a quantity which could be modelled as
   1. a discrete random variable,
   2. a continuous random variable.
   [2 marks]

The discrete random variable \(X\) has the probability function given by the following table:

1. Show that \(p = 0.07\) [2 marks]
2. Find the value of \(E(X + 2)\). [4 marks]
3. Find the value of \(\text{Var}(3X - 1)\). [5 marks]

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

1. the product-moment correlation coefficient between \(x\) and \(y\), [3 marks]
2. the equation of the regression line of \(y\) on \(x\). [4 marks]

Given that \(p = x + 30\) and \(q = y + 50\),

1. find the equation of the regression line of \(q\) on \(p\), in the form \(q = mp + c\). [3 marks]
2. Estimate the value of \(q\) when \(p = 46\), stating any assumptions you make. [3 marks]

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, [9 marks]
2. the percentage of the students who are under 158 cm tall. [3 marks]
3. Comment briefly on the suitability of a normal distribution to model such a population. [2 marks]

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. [8 marks]
2. Find the three quartiles for the distribution. [4 marks]
3. On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers. [4 marks]

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 $$r(r - 1) = 420.$$ [4 marks]
2. Hence or otherwise find the value of \(r\). [3 marks]
3. Find the probability that, when three cards are drawn at random from the pack,
   1. at least two are red, [6 marks]
   2. the first one is red given that at least two are red. [4 marks]