CAIE S2 (Statistics 2) 2018 June

1 A random variable \(X\) has the distribution \(\mathrm { B } ( 75,0.03 )\).

1. Use the Poisson approximation to the binomial distribution to calculate \(\mathrm { P } ( X < 3 )\).
2. Justify the use of the Poisson approximation.

2 Amy has to choose a random sample from the 265 students in her year at college. She numbers the students from 1 to 265 and then uses random numbers generated by her calculator. The first two random numbers produced by her calculator are 0.213165448 and 0.073165196 .

1. Use these figures to find the numbers of the first four students in her sample.
   There were 25 students in Amy's sample. She asked each of them how much money, \(
   ) x$, they earned in a week, on average. Her results are summarised below. $$n = 25 \quad \Sigma x = 510 \quad \Sigma x ^ { 2 } = 13225$$
2. Find unbiased estimates of the population mean and variance.
3. Explain briefly what is meant by 'population' in this question.

3 A researcher wishes to estimate the proportion, \(p\), of houses in London Road that have only one occupant. He takes a random sample of 64 houses in London Road and finds that 8 houses in the sample have only one occupant. Using this sample, he calculates that an approximate \(\alpha \%\) confidence interval for \(p\) has width 0.130 . Find \(\alpha\) correct to the nearest integer.

4 The numbers, \(M\) and \(F\), of male and female students who leave a particular school each year to study engineering have means 3.1 and 0.8 respectively.

1. State, in context, one condition required for \(M\) to have a Poisson distribution.
   Assume that \(M\) and \(F\) can be modelled by independent Poisson distributions.
2. Find the probability that the total number of students who leave to study engineering in a particular year is more than 3 .
3. Given that the total number of students who leave to study engineering in a particular year is more than 3 , find the probability that no female students leave to study engineering in that year.

5 The time taken for a particular train journey is normally distributed. In the past, the time had mean 2.4 hours and standard deviation 0.3 hours. A new timetable is introduced and on 30 randomly chosen occasions the time for this journey is measured. The mean time for these 30 occasions is found to be 2.3 hours.

1. Stating any assumption(s), test, at the \(5 \%\) significance level, whether the mean time for this journey has changed.
2. A similar test at the \(5 \%\) significance level was carried out using the times from another randomly chosen 30 occasions.
   (a) State the probability of a Type I error.
   (b) State what is meant by a Type II error in this context.

6 The times, in minutes, taken to complete the two parts of a task are normally distributed with means 4.5 and 2.3 respectively and standard deviations 1.1 and 0.7 respectively.

1. Find the probability that the total time taken for the task is less than 8.5 minutes.
2. Find the probability that the time taken for the first part of the task is more than twice the time taken for the second part.

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

1. Show that \(k = \frac { 8 } { 7 }\).
2. Find \(\mathrm { E } ( X )\).
3. Three values of \(X\) are chosen at random. Find the probability that one of these values is less than 1.5 and the other two are greater than 1.5.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.