Edexcel S2 (Statistics 2) 2015 June

In a survey it is found that barn owls occur randomly at a rate of 9 per 1000 km\(^2\).

1. Find the probability that in a randomly selected area of 1000 km\(^2\) there are at least 10 barn owls. [2]
2. Find the probability that in a randomly selected area of 200 km\(^2\) there are exactly 2 barn owls. [3]
3. Using a suitable approximation, find the probability that in a randomly selected area of 50000 km\(^2\) there are at least 470 barn owls. [6]

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.

1. Find P(5 \(\leq X < 11\)) [3]

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.

1. Test, at the 10\% level of significance, the radio company's claim. State your hypotheses clearly. [5]

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

1. Show that \(k = \frac{1}{4}\) [4]
2. Write down the mode of \(X\). [1]
3. Specify fully the cumulative distribution function F(\(x\)). [5]
4. Find the upper quartile of \(X\). [4]

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 P(\(L < 0.4\)). [1]
2. Write down E(\(L\)). [1]
3. Calculate Var(\(L\)). [2]

A random sample of 30 poles cut by this machine is taken.

1. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. [3]

When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function F(\(x\)) where $$\text{F}(x) = \begin{cases} 0 & x < 0 \\ 4x - 4x^2 & 0 \leq x \leq 0.5 \\ 1 & \text{otherwise} \end{cases}$$

1. Using this model, find P(\(X > 0.4\)) [2]

A random sample of 100 poles cut by this new machine is taken.

1. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres. [3]

\emph{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. [3]

Over a randomly selected 1 month period the lift broke down 3 times.

1. Test, at the 5\% level of significance, whether \emph{Liftsforall}'s claim is correct. State your hypotheses clearly. [2]
2. State the actual significance level of this test. [1]

The residents in the block of flats have a maintenance contract with \emph{Liftsforall}. The residents pay \emph{Liftsforall} £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. \emph{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,

1. find the probability that the residents do not pay more than £500 to \emph{Liftsforall} in the next year. [6]

A continuous random variable \(X\) has probability density function f(\(x\)) where $$f(x) = \begin{cases} kx^n & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(n\) are positive integers.

1. Find \(k\) in terms of \(n\). [3]
2. Find E(\(X\)) in terms of \(n\). [3]
3. Find E(\(X^2\)) in terms of \(n\). [2]

Given that \(n = 2\)

1. find Var(3\(X\)). [3]

A bag contains a large number of 10p, 20p and 50p 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. [7]