OCR S2 (Statistics 2) 2010 June

1

1. The number of inhabitants of a village who are selected for jury service in the course of a 10-year period is a random variable with the distribution \(\operatorname { Po } ( 4.2 )\).
   (a) Find the probability that in the course of a 10-year period, at least 7 inhabitants are selected for jury service.
   (b) Find the probability that in 1 year, exactly 2 inhabitants are selected for jury service.
2. Explain why the number of inhabitants of the village who contract influenza in 1 year can probably not be well modelled by a Poisson distribution.

2 A university has a large number of students, of whom \(35 \%\) are studying science subjects. A sample of 10 students is obtained by listing all the students, giving each a serial number and selecting by using random numbers.

1. Find the probability that fewer than 3 of the sample are studying science subjects.
2. It is required that, in selecting the sample, the same student is not selected twice. Explain whether this requirement invalidates your calculation in part (i).

3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).

1. Find the value of \(\sigma\).
2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).

4 The proportion of commuters in a town who travel to work by train is 0.4 . Following the opening of a new station car park, a random sample of 16 commuters is obtained, and 11 of these travel to work by train. Test at the \(1 \%\) significance level whether there is evidence of an increase in the proportion of commuters in this town who travel to work by train.

5 The time \(T\) seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.

1. A test is carried out, at the \(5 \%\) significance level, of whether the specification is being met, using the mean \(\bar { t }\) of a random sample of 10 times.
   (a) Find the critical region for the test, in terms of \(\bar { t }\).
   (b) Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
2. Because of system degradation and memory load, the population mean time \(\mu\) seconds increases with the number of months of use, \(m\). A formula for \(\mu\) in terms of \(m\) is \(\mu = 20 + 0.6 m\). Use this formula to find the value of \(m\) for which the probability that the test results in rejection of the null hypothesis is 0.5 .

6

1. The random variable \(D\) has the distribution \(\operatorname { Po } ( 24 )\). Use a suitable approximation to find \(P ( D > 30 )\).
2. An experiment consists of 200 trials. For each trial, the probability that the result is a success is 0.98 , independent of all other trials. The total number of successes is denoted by \(E\).
   1. Explain why the distribution of \(E\) cannot be well approximated by a Poisson distribution.
   2. By considering the number of failures, use an appropriate Poisson approximation to find \(\mathrm { P } ( E \leqslant 194 )\).

7 A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, \(x\) micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the \(10 \%\) significance level, whether the machine is producing paper of the designed thickness.

8 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k x ^ { - a } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants and \(a\) is greater than 1 .

1. Show that \(k = a - 1\).
2. Find the variance of \(X\) in the case \(a = 4\).
3. It is given that \(\mathrm { P } ( X < 2 ) = 0.9\). Find the value of \(a\), correct to 3 significant figures.