AQA S2 (Statistics 2) 2006 June

1 The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

1. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
2. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
   1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
   2. What assumption did you make in answering part (b)(i)?

2 The weights of lions kept in captivity at Wildcat Safari Park are normally distributed.
The weights, in kilograms, of a random sample of five lions were recorded as $$\begin{array} { l l l l l } 46 & 48 & 57 & 49 & 54 \end{array}$$

1. Construct a 95\% confidence interval for the mean weight of lions kept in captivity at Wildcat Safari Park.
2. State the probability that this confidence interval does not contain the mean weight of lions kept in captivity at Wildcat Safari Park.

3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, \(R\), scored in each game by the team can be modelled by the following probability distribution.

1. Calculate exact values for the mean and variance of \(R\).
2. Next season the team will play 32 games. They expect to win \(90 \%\) of the games in which they score at least three goals, half of the games in which they score exactly two goals and \(20 \%\) of the games in which they score exactly one goal. Find, for next season:
   1. the number of games in which they expect to score at least three goals;
   2. the number of games that they expect to win.

4 It is claimed that the area within which a school is situated affects the age profile of the staff employed at that school. In order to investigate this claim, the age profiles of staff employed at two schools with similar academic achievements are compared. Academia High School, situated in a rural community, employs 120 staff whilst Best Manor Grammar School, situated in an inner-city community, employs 80 staff. The percentage of staff within each age group, for each school, is given in the table.

1. 1. Form the data into a contingency table suitable for analysis using a \(\chi ^ { 2 }\) distribution.
      (2 marks)
   2. Use a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to determine whether there is an association between the age profile of the staff employed and the area within which the school is situated.
2. Interpret your result in part (a)(ii) as it relates to the 22-34 age group.

5

1. The continuous random variable \(X\) follows a rectangular distribution with probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b
   0 & \text { otherwise } \end{cases}$$
   1. Write down \(\mathrm { E } ( X )\).
   2. Prove, using integration, that $$\operatorname { Var } ( X ) = \frac { 1 } { 12 } b ^ { 2 }$$
2. At an athletics meeting, the error, in seconds, made in recording the time taken to complete the 10000 metres race may be modelled by the random variable \(T\), having the probability density function $$f ( t ) = \left\{ \begin{array} { c c } 5 & - 0.1 \leqslant t \leqslant 0.1
   0 & \text { otherwise } \end{array} \right.$$ Calculate \(\mathrm { P } ( | T | > 0.02 )\).

6 The lifetime, \(X\) hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.

1. The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows: $$\begin{array} { l l l l l } 85 & 92 & 100 & 95 & 99 \end{array}$$ Test the members' suspicion at the \(5 \%\) level of significance.
2. The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, \(x\) hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where \(\bar { x }\) denotes the sample mean: $$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$ Test the manufacturer's belief at the \(5 \%\) level of significance.

7 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1
\frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. 1. Show that the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leqslant x \leqslant 1\) is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
   2. Hence write down the value of \(\mathrm { P } ( X \leqslant 1 )\).
   3. Find the value of \(x\) for which \(\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }\).
   4. Find the lower quartile of the distribution.