Edexcel S2 (Statistics 2) 2017 October

1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)

The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .

1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
   

2. The weekly sales, \(S\), in thousands of pounds, of a small business has probability density function $$\mathrm { f } ( s ) = \left\{ \begin{array} { c c } k ( s - 2 ) ( 10 - s ) & 2 < s < 10
0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to show that \(k = \frac { 3 } { 256 }\)
2. Write down the value of \(\mathrm { E } ( S )\)
3. Use algebraic integration to find the standard deviation of the weekly sales. A week is selected at random.
4. Showing your working, find the probability that this week's sales exceed \(\pounds 7100\) Give your answer to one decimal place. A quarter is defined as 12 consecutive weeks. The discrete random variable \(X\) is the number of weeks in a quarter in which the weekly sales exceed £7100 The manager earns a bonus at the following rates:

   \(\boldsymbol { X }\)	Bonus Earned
   \(X \leqslant 5\)	\(\pounds 0\)
   \(X = 6\)	\(\pounds 1000\)
   \(X \geqslant 7\)	\(\pounds 5000\)

5. Using your answer to part (d), calculate the manager's expected bonus per quarter.

3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8 The shop has 9 Birdscope cameras in stock at the start of each week. A week is selected at random.

1. Find the probability that the demand for Birdscope cameras cannot be met in this particular week. In a year, there are 50 weeks in which Birdscope cameras can be sold.
2. Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
3. Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2 The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold. Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
4. use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.

4. In a computer game, a ship appears randomly on a rectangular screen. The continuous random variable \(X \mathrm {~cm}\) is the distance of the centre of the ship from the bottom of the screen. The random variable \(X\) is uniformly distributed over the interval \([ 0 , \alpha ]\) where \(\alpha \mathrm { cm }\) is the height of the screen. Given that \(\mathrm { P } ( X > 6 ) = 0.6\)

1. find the value of \(\alpha\)
2. find \(\mathrm { P } ( 4 < X < 10 )\) The continuous random variable \(Y\) cm is the distance of the centre of the ship from the left-hand side of the screen. The random variable \(Y\) is uniformly distributed over the interval [ 0,20 ] where 20 cm is the width of the screen.
3. Find the mean and the standard deviation of \(Y\).
4. Find \(\mathrm { P } ( | Y - 4 | < 2 )\)
5. Given that \(X\) and \(Y\) are independent, find the probability that the centre of the ship appears
   1. in a square of side 4 cm which is at the centre of the screen,
   2. within 5 cm of a side or the top or the bottom of the screen.

5. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 3
k \left( y ^ { 2 } - 2 y - 3 \right) & 3 \leqslant y \leqslant \alpha
4 k ( 2 y - 7 ) & \alpha < y \leqslant 6
1 & y > 6 \end{array} \right.$$ where \(k\) and \(\alpha\) are constants.

1. Find \(\mathrm { P } ( 4.5 < Y \leqslant 5.5 )\)
2. Find the probability density function \(\mathrm { f } ( \mathrm { y } )\)

6. A fair 6 -sided die is thrown \(n\) times. The number of sixes, \(X\), is recorded. Using a normal approximation, \(\mathrm { P } ( X < 50 ) = 0.0082\) correct to 4 decimal places. Find the value of \(n\).
(10)