OCR S2 (Statistics 2) 2012 January

1 A random sample of 50 observations of the random variable \(X\) is summarised by $$n = 50 , \Sigma x = 182.5 , \Sigma x ^ { 2 } = 739.625 .$$ Calculate unbiased estimates of the expectation and variance of \(X\).

2 The random variable \(Y\) has the distribution \(\mathrm { B } ( 140,0.03 )\). Use a suitable approximation to find \(\mathrm { P } ( Y = 5 )\). Justify your approximation.

3 The random variable \(G\) has a normal distribution. It is known that $$\mathrm { P } ( G < 56.2 ) = \mathrm { P } ( G > 63.8 ) = 0.1 \text {. }$$ Find \(\mathrm { P } ( G > 65 )\).

4 The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that \(\mathrm { E } ( H ) = 2.5\) and \(\operatorname { Var } ( H ) = 1.25\). The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar { H }\).
Use a suitable approximation to find \(\mathrm { P } ( \bar { H } < 2.6 )\).

5

1. Six prizes are allocated, using random numbers, to a group of 12 girls and 8 boys. Calculate the probability that exactly 4 of the prizes are allocated to girls if
   (a) the same child may win more than one prize,
   (b) no child may win more than one prize.
2. Sixty prizes are allocated, using random numbers, to a group of 1200 girls and 800 boys. Use a suitable approximation to calculate the probability that at least 30 of the prizes are allocated to girls. Does it affect your calculation whether or not the same child may win more than one prize? Justify your answer.

6 The number of fruit pips in 1 cubic centimetre of raspberry jam has the distribution \(\operatorname { Po } ( \lambda )\). Under a traditional jam-making process it is known that \(\lambda = 6.3\). A new process is introduced and a random sample of 1 cubic centimetre of jam produced by the new process is found to contain 2 pips. Test, at the \(5 \%\) significance level, whether this is evidence that under the new process the average number of pips has been reduced. Find (a) \(\mathrm { E } ( X )\),
(ii) The continuous random variable \(Y\) has the probability density function $$g ( y ) = \left\{ \begin{array} { l r } \frac { 1.5 } { y ^ { 2.5 } } & y \geqslant 1
0 & \text { otherwise. } \end{array} \right.$$ Given that \(\mathrm { E } ( Y ) = 3\), show that \(\operatorname { Var } ( Y )\) is not finite.

8 In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.

1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. Assume now that a Poisson model is appropriate.
2. Find the probability that in 10 ml there are at least 5 bacteria.
3. Find the probability that in 3.7 ml there are exactly 2 bacteria.
4. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation.

9 It is desired to test whether the average amount of sleep obtained by school pupils in Year 11 is 8 hours, based on a random sample of size 64. The population standard deviation is 0.87 hours and the sample mean is denoted by \(\bar { H }\). The critical values for the test are \(\bar { H } = 7.72\) and \(\bar { H } = 8.28\).

1. State appropriate hypotheses for the test, explaining the meaning of any symbol you use.
2. Calculate the significance level of the test.
3. Explain what is meant by a Type I error in this context.
4. Given that in fact the average amount of sleep obtained by all pupils in Year 11 is 7.9 hours, find the probability that the test results in a Type II error. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}