OCR S2 (Statistics 2) Specimen

1 The standard deviation of a random variable \(F\) is 12.0. The mean of \(n\) independent observations of \(F\) is denoted by \(\bar { F }\).

1. Given that the standard deviation of \(\bar { F }\) is 1.50 , find the value of \(n\).
2. For this value of \(n\), state, with justification, what can be said about the distribution of \(\bar { F }\).

2 A certain neighbourhood contains many small houses (with small gardens) and a few large houses (with large gardens). A sample survey of all houses is to be carried out in this neighbourhood. A student suggests that the sample could be selected by sticking a pin into a map of the neighbourhood the requisite number of times, while blindfolded.

1. Give two reasons why this method does not produce a random sample.
2. Describe a better method.

3 Sixty people each make two throws with a fair six-sided die.

1. State the probability of one particular person obtaining two sixes.
2. Using a suitable approximation, calculate the probability that at least four of the sixty obtain two sixes.

4 The random variable \(G\) has mean 20.0 and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( G > 15.0 ) = 0.6\). Assume that \(G\) is normally distributed.

1. (a) Find the value of \(\sigma\).
   (b) Given that \(\mathrm { P } ( G > g ) = 0.4\), find the value of \(\mathrm { P } ( G > 2 g )\).
2. It is known that no values of \(G\) are ever negative. State with a reason what this tells you about the assumption that \(G\) is normally distributed.

5 The mean solubility rating of widgets inserted into beer cans is thought to be 84.0, in appropriate units. A random sample of 50 widgets is taken. The solubility ratings, \(x\), are summarised by $$n = 50 , \quad \Sigma x = 4070 , \quad \Sigma x ^ { 2 } = 336100$$ Test, at the \(5 \%\) significance level, whether the mean solubility rating is less than 84.0 .

6 On average a motorway police force records one car that has run out of petrol every two days.

1. (a) Using a Poisson distribution, calculate the probability that, in one randomly chosen day, the police force records exactly two cars that have run out of petrol.
   (b) Using a Poisson distribution and a suitable approximation to the binomial distribution, calculate the probability that, in one year of 365 days, there are fewer than 205 days on which the police force records no cars that have run out of petrol.
2. State an assumption needed for the Poisson distribution to be appropriate in part (i), and explain why this assumption is unlikely to be valid.

7 The time, in minutes, for which a customer is prepared to wait on a telephone complaints line is modelled by the random variable \(X\). The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 81 }\).
2. Find \(\mathrm { E } ( X )\).
3. (a) Show that the value \(y\) which satisfies \(\mathrm { P } ( X < y ) = \frac { 3 } { 5 }\) satisfies $$5 y ^ { 4 } - 90 y ^ { 2 } + 243 = 0 .$$ (b) Using the substitution \(w = y ^ { 2 }\), or otherwise, solve the equation in part (a) to find the value of \(y\).

8 The proportion of left-handed adults in a country is known to be \(15 \%\). It is suggested that for mathematicians the proportion is greater than \(15 \%\). A random sample of 12 members of a university mathematics department is taken, and it is found to include five who are left-handed.

1. Stating your hypotheses, test whether the suggestion is justified, using a significance level as close to \(5 \%\) as possible.
2. In fact the significance test cannot be carried out at a significance level of exactly \(5 \%\). State the probability of making a Type I error in the test.
3. Find the probability of making a Type II error in the test for the case when the proportion of mathematicians who are left-handed is actually \(20 \%\).
4. Determine, as accurately as the tables of cumulative binomial probabilities allow, the actual proportion of mathematicians who are left-handed for which the probability of making a Type II error in the test is 0.01 .