Edexcel S1 (Statistics 1) 2004 January

1. An office has the heating switched on at 7.00 a.m. each morning. On a particular day, the temperature of the office, \(t { } ^ { \circ } \mathrm { C }\), was recorded \(m\) minutes after 7.00 a.m. The results are shown in the table below.

1. Calculate the exact values of \(S _ { m t }\) and \(S _ { m m }\).
2. Calculate the equation of the regression line of \(t\) on \(m\) in the form \(t = a + b m\).
3. Use your equation to estimate the value of \(t\) at 7.35 a.m.
4. State, giving a reason, whether or not you would use the regression equation in (b) to estimate the temperature
   1. at 9.00 a.m. that day,
   2. at 7.15 a.m. one month later.

2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).

1. Write down 3 properties of the distribution of \(X\). Given that \(\mu = 27\) and \(\sigma = 10\)
2. find \(\mathrm { P } ( 26 < X < 28 )\).

3. A discrete random variable \(X\) has the probability function shown in the table below. Find

1. \(\mathrm { P } ( 1 < X \leq 3 )\),
2. \(\mathrm { F } ( 2.6 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 2 X - 3 )\),
5. \(\operatorname { Var } ( X )\)

4. \(\quad\) The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }\) and \(\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }\).

1. Find
   1. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
   2. \(\mathrm { P } ( A \cap B )\),
   3. \(\mathrm { P } ( A \cup B )\),
   4. \(\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)\).
2. State, with a reason, whether or \(\operatorname { not } A\) and \(B\) are
   1. mutually exclusive,
   2. independent.

5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.

1. Draw a histogram to represent these data.
2. Use interpolation to estimate the median and inter-quartile range of daily sales.
3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
   (2)

6. One of the objectives of a computer game is to collect keys. There are three stages to the game. The probability of collecting a key at the first stage is \(\frac { 2 } { 3 }\), at the second stage is \(\frac { 1 } { 2 }\), and at the third stage is \(\frac { 1 } { 4 }\).

1. Draw a tree diagram to represent the 3 stages of the game.
2. Find the probability of collecting all 3 keys.
3. Find the probability of collecting exactly one key in a game.
4. Calculate the probability that keys are not collected on at least 2 successive stages in a game.