Edexcel S2 (Statistics 2) 2015 June

1. In a survey it is found that barn owls occur randomly at a rate of 9 per \(1000 \mathrm {~km} ^ { 2 }\).
   1. Find the probability that in a randomly selected area of \(1000 \mathrm {~km} ^ { 2 }\) there are at least 10 barn owls.
   2. Find the probability that in a randomly selected area of \(200 \mathrm {~km} ^ { 2 }\) there are exactly 2 barn owls.
   3. Using a suitable approximation, find the probability that in a randomly selected area of \(50000 \mathrm {~km} ^ { 2 }\) there are at least 470 barn owls.
   4. The proportion of houses in Radville which are unable to receive digital radio is \(25 \%\). In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.
   5. Find \(\mathrm { P } ( 5 \leqslant X < 11 )\)
   A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.
2. Test, at the \(10 \%\) level of significance, the radio company's claim. State your hypotheses clearly.

3. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 2
k \left( 1 - \frac { x } { 6 } \right) & 2 < x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 4 }\)
2. Write down the mode of \(X\).
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find the upper quartile of \(X\).

1. The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval \([ 0,0.5 ]\)
   1. Find \(\mathrm { P } ( L < 0.4 )\).
   2. Write down \(\mathrm { E } ( L )\).
   3. Calculate \(\operatorname { Var } ( L )\).
   A random sample of 30 poles cut by this machine is taken.
2. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
   4 x - 4 x ^ { 2 } & 0 \leqslant x \leqslant 0.5
   1 & \text { otherwise } \end{array} \right.$$
3. Using this model, find \(\mathrm { P } ( X > 0.4 )\) A random sample of 100 poles cut by this new machine is taken.
4. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.

1. Liftsforall claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.
   1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to \(2.5 \%\) as possible.
   Over a randomly selected 1 month period the lift broke down 3 times.
2. Test, at the \(5 \%\) level of significance, whether Liftsforall's claim is correct. State your hypotheses clearly.
3. State the actual significance level of this test.
   ! The residents in the block of flats have a maintenance contract with Liftsforall. The residents pay Liftsforall \(\pounds 500\) for every quarter ( 3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. Liftsforall installs a new lift in the block of flats.
   Given that the new lift breaks down at a mean rate of 2 times per month,
4. find the probability that the residents do not pay more than \(\pounds 500\) to Liftsforall in the next year.

1. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where

$$f ( x ) = \left\{ \begin{array} { c c } k x ^ { n } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(n\) are positive integers.

1. Find \(k\) in terms of \(n\).
2. Find \(\mathrm { E } ( X )\) in terms of \(n\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\) in terms of \(n\). Given that \(n = 2\)
4. find \(\operatorname { Var } ( 3 X )\).

1. A bag contains a large number of \(10 \mathrm { p } , 20 \mathrm { p }\) and 50 p coins in the ratio \(1 : 2 : 2\)

A random sample of 3 coins is taken from the bag.
Find the sampling distribution of the median of these samples.