AQA S2 (Statistics 2) 2008 January

1 David claims that customers have to queue at a supermarket checkout for more than 5 minutes, on average. The queuing times, \(x\) minutes, of 40 randomly selected customers result in \(\bar { x } = 5.5\) and \(s ^ { 2 } = 1.31\). Investigate, at the \(1 \%\) level of significance, David's claim.

2 A new information technology centre is advertising places on its one-week residential computer courses.

1. The number of places, \(X\), booked each week on the publishing course may be modelled by a Poisson distribution with a mean of 9.0.
   1. State the standard deviation of \(X\).
   2. Calculate \(\mathrm { P } ( 6 < X < 12 )\).
2. The number of places booked each week on the web design course may be modelled by a Poisson distribution with a mean of 2.5.
   1. Write down the distribution for \(T\), the total number of places booked each week on the publishing and web design courses.
   2. Hence calculate the probability that, during a given week, a total of fewer than 2 places are booked.
3. The number of places booked on the database course during each of a random sample of 10 weeks is as follows: $$\begin{array} { l l l l l l l l l l } 14 & 15 & 8 & 16 & 18 & 4 & 10 & 12 & 15 & 8 \end{array}$$ By calculating appropriate numerical measures, state, with a reason, whether or not the Poisson distribution \(\mathrm { Po } ( 12.0 )\) could provide a suitable model for the number of places booked each week on the database course.

3

1. The continuous random variable \(T\) follows a rectangular distribution with probability density function given by $$\mathrm { f } ( t ) = \left\{ \begin{array} { l c } k & - a \leqslant t \leqslant b
   0 & \text { otherwise } \end{array} \right.$$
   1. Express \(k\) in terms of \(a\) and \(b\).
   2. Prove, using integration, that \(\mathrm { E } ( T ) = \frac { 1 } { 2 } ( b - a )\).
2. The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 10 } & - 4 \leqslant t \leqslant 6
   0 & \text { otherwise } \end{array} \right.$$
   1. Write down the value of \(\mathrm { E } ( T )\).
   2. Calculate \(\mathrm { P } ( T < - 3\) or \(T > 3 )\).

4 A speed camera was used to measure the speed, \(V\) mph, of John's serves during a tennis singles championship. For 10 randomly selected serves, $$\sum v = 1179 \quad \text { and } \quad \sum ( v - \bar { v } ) ^ { 2 } = 1014.9$$ where \(\bar { v }\) is the sample mean.

1. Construct a \(99 \%\) confidence interval for the mean speed of John's serves at this tennis championship, stating any assumption that you make.
   (7 marks)
2. Hence comment on John's claim that, at this championship, he consistently served at speeds in excess of 130 mph .
   (1 mark)

5 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { x } { 20 } & x = 1,2,3,4,5
\frac { x } { 24 } & x = 6
0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X \geqslant 5 )\).
2. 1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }\).
   2. Hence, or otherwise, show that \(\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036\), correct to three decimal places.
3. Calculate the mean and the variance of \(A\), the area of rectangles having sides of length \(X + 3\) and \(\frac { 1 } { X }\).

6 A survey is carried out in an attempt to determine whether the salary achieved by the age of 30 is associated with having had a university education. The results of this survey are given in the table.

1. Use a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to determine whether the salary achieved by the age of 30 is associated with having had a university education.
2. What do you understand by a Type I error in this context?

7 The waiting time, \(X\) minutes, for fans to gain entrance to see an event may be modelled by a continuous random variable having the distribution function defined by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4
1 & x > 4 \end{cases}$$

1. 1. Sketch the graph of F.
   2. Explain why the value of \(q _ { 1 }\), the lower quartile of \(X\), is \(\frac { 1 } { 2 }\).
   3. Show that the upper quartile, \(q _ { 3 }\), satisfies \(1.6 < q _ { 3 } < 1.7\).
2. The probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1
   \beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4
   0 & \text { otherwise } \end{cases}$$
   1. Show that the exact values of \(\alpha\) and \(\beta\) are \(\frac { 1 } { 2 }\) and \(\frac { 1 } { 18 }\) respectively.
   2. Hence calculate \(\mathrm { E } ( X )\).