Edexcel S2 (Statistics 2) 2011 June

1. A factory produces components. Each component has a unique identity number and it is assumed that \(2 \%\) of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.
   1. Identify a sampling frame.
   The statistic \(F\) represents the number of faulty components in the random sample of size 50.
2. Specify the sampling distribution of \(F\).

2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.

1. Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval. The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.
3. Using a \(5 \%\) level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\).
For \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant. For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ). For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find

1. the mode,
2. the value of \(k\),
3. the value of \(a\). Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
4. state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).

1. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].

A stick is selected at random from the box.

1. Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
2. Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
3. Find the probability of winning a soft toy.

5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .

1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.

1. A shopkeeper knows, from past records, that \(15 \%\) of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.
   1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.
   During the refurbishment a new sandwich display was installed. Before the refurbishment \(20 \%\) of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.
2. Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper's claim. Use a \(10 \%\) level of significance.

1. The continuous random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 3 } { 32 } ( x - 1 ) ( 5 - x ) & 1 \leqslant x \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

1. Sketch \(\mathrm { f } ( x )\) showing clearly the points where it meets the \(x\)-axis.
2. Write down the value of the mean, \(\mu\), of \(X\).
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 9.8\)
4. Find the standard deviation, \(\sigma\), of \(X\). The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
   \frac { 1 } { 32 } \left( a - 15 x + 9 x ^ { 2 } - x ^ { 3 } \right) & 1 \leqslant x \leqslant 5
   1 & x > 5 \end{array} \right.$$ where \(a\) is a constant.
5. Find the value of \(a\).
6. Show that the lower quartile of \(X , q _ { 1 }\), lies between 2.29 and 2.31
7. Hence find the upper quartile of \(X\), giving your answer to 1 decimal place.
8. Find, to 2 decimal places, the value of \(k\) so that $$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$