OCR S2 (Statistics 2) 2016 June

1 The results of 14 observations of a random variable \(V\) are summarised by $$n = 14 , \quad \sum v = 3752 , \quad \sum v ^ { 2 } = 1007448 .$$ Calculate unbiased estimates of \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).

2 The mass, in kilograms, of a packet of flour is a normally distributed random variable with mean 1.03 and variance \(\sigma ^ { 2 }\). Given that \(5 \%\) of packets have mass less than 1.00 kg , find the percentage of packets with mass greater than 1.05 kg .

3 The random variable \(F\) has the distribution \(\mathrm { B } ( 40,0.65 )\). Use a suitable approximation to find \(\mathrm { P } ( F \leqslant 30 )\), justifying your approximation.

4 It is given that \(Y \sim \operatorname { Po } ( \lambda )\), where \(\lambda \neq 0\), and that \(\mathrm { P } ( Y = 4 ) = \mathrm { P } ( Y = 5 )\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of \(\mathrm { P } ( Y = 5 )\).
\(555 \%\) of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55 . As evidence for his claim he says that 6 of the last 8 Head Students have been girls.

1. Use an exact binomial distribution to test the claim at the \(10 \%\) significance level.
2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid.

6 The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution.
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate \(\mathrm { P } ( 4 \leqslant X < 8 )\).
3. At a busy time of the day, \(\lambda = 30\).
   (a) Assuming that a Poisson distribution is valid, use a suitable approximation to find \(\mathrm { P } ( X > 35 )\). Justify your approximation.
   (b) Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } a x ^ { - 3 } + b x ^ { - 4 } & x \geqslant 1
0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. Explain what the letter \(x\) represents. It is given that \(\mathrm { P } ( X > 2 ) = \frac { 3 } { 16 }\).
2. Show that \(a = 1\), and find the value of \(b\).
3. Find \(\mathrm { E } ( X )\).

8 It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years \(^ { 2 }\).

1. Test at the \(5 \%\) significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild.
2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years \({ } ^ { 2 }\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test.
3. Explain whether the Central Limit Theorem is needed in these tests.

9 The random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out at the \(1 \%\) level of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 11\) against \(\mathrm { H } _ { 1 } : \lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14 , find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. \section*{END OF QUESTION PAPER}