Edexcel S2 (Statistics 2)

1. (a) Explain why it is often useful to take samples as a means of obtaining information.
   (b) Briefly define the term sampling frame.
   (c) Suggest a suitable sampling frame for a sample survey on a proposal to install speed humps on a road.
2. An insurance company conducts its business by using a Call Centre. The average number of calls per minute is \(3 \cdot 5\). In the first minute after a TV advertisement is shown, the number of calls received is 7 .
   (a) Stating your hypotheses carefully, and working at the \(5 \%\) significance level, test whether the advertisement has had an effect.
   (b) Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the \(0.1 \%\) significance level.
3. On average, \(35 \%\) of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that
   (a) less than 5 get A or B grades,
   (b) exactly 8 get A or B grades.

Five such classes of 20 students are combined to sit the exam.
(c) Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades.

4. Light bulbs produced in a certain factory have lifetimes, in 100 s of hours, whose distribution is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x ( 3 - x ) } { 9 } , & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Write down the mean lifetime of a bulb.
3. Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours.
4. Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours.
5. State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs. \section*{STATISTICS 2 (A) TEST PAPER 2 Page 2}

1. In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit.
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. Another machine produces biscuits with a mean of 1.9 currants per biscuit.
3. Determine which machine is more likely to produce a biscuit with at least two currants.

6. A greengrocer sells apples from a barrel in his shop. He claims that no more than \(5 \%\) of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.

1. Stating your hypotheses clearly, test his claim at the \(1 \%\) significance level.
2. State an assumption that has been made about the selection of the apples.
3. When five other customers also buy 10 apples each, the numbers of bad apples they get are \(1,3,1,2\) and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the \(1 \%\) significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than \(5 \%\).
4. Comment briefly on your results in parts (a) and (c).

7. Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.

1. Find the mean and the standard deviation of \(X\). It is suggested that a better model would be the distribution with probability density function $$f ( x ) = c x , 0 \leq x \leq 5 , \quad f ( x ) = c ( 10 - x ) , 5 < x \leq 10 , \quad f ( x ) = 0 \text { otherwise. }$$
2. Write down the mean of \(X\).
3. Find \(c\), and hence find the standard deviation of \(X\) in this model.
4. Find \(\mathrm { P } ( 4 < X < 6 )\). It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
5. Use these results to find \(\mathrm { P } ( 4 < X < 6 )\) in the third model.
6. Compare your answer with (d). Which model do you think is most appropriate? (1 mark)