OCR S3 (Statistics 3) 2011 June

1 The random variables \(X\) and \(Y\) are independent with \(X \sim \operatorname { Po } ( 5 )\) and \(Y \sim \operatorname { Po } ( 4 )\). \(S\) denotes the sum of 2 observations of \(X\) and 3 observations of \(Y\).

1. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. The random variable \(T\) is defined by \(\frac { 1 } { 2 } X - \frac { 1 } { 4 } Y\). Show that \(\mathrm { E } ( T ) = \operatorname { Var } ( T )\).
3. State which of \(S\) and \(T\) (if either) does not have a Poisson distribution, giving a reason for your answer.

2 The population proportion of all men with red-green colour blindness is denoted by \(p\). Each of a random sample of 80 men was tested and it was found that 6 had red-green colour blindness.

1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
2. For a different random sample of men, the proportion with red-green colour blindness is denoted by \(p _ { s }\). Estimate the sample size required in order that \(\left| p _ { s } - p \right| \leqslant 0.05\) with probability \(95 \%\).
3. Give one reason why the calculated sample size is an estimate.

3 The monthly demand for a product, \(X\) thousand units, is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1
a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find

1. the value of \(a\),
2. the probability that the monthly demand is at most 1500 units,
3. the expected monthly demand.

4 An experiment by Lord Rutherford at Cambridge in 1909 involved measuring the numbers of \(\alpha\)-particles emitted during radioactive decay. The following table shows emissions during 2608 intervals of 7.5 seconds. It is given that the mean number of particles emitted per interval, calculated from the data, is 3.87 , correct to 3 significant figures.

1. Find the contribution to the \(\chi ^ { 2 }\) value of the frequency of 273 corresponding to \(x = 6\) in a goodness of fit test for a Poisson distribution.
2. Given that no cells need to be combined, state why the number of degrees of freedom is 10 .
3. Given also that the calculated value of \(\chi ^ { 2 }\) is 13.0 , correct to 3 significant figures, carry out the test at the 10\% significance level.

5 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 4 } { 3 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 2 ,
1 & x > 2 . \end{cases}$$

1. Find the median value of \(X\).
2. Find the (cumulative) distribution function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), and hence find the probability density function of \(Y\).
3. Evaluate \(\mathrm { E } \left( 2 - \frac { 2 } { X ^ { 2 } } \right)\).

6 The Body Mass Index (BMI) of each of a random sample of 100 army recruits from a large intake in 2008 was measured. The results are summarised by $$\Sigma x = 2605.0 , \quad \Sigma x ^ { 2 } = 68636.41 .$$ It may be assumed that BMI has a normal distribution.

1. Find a 98\% confidence interval for the mean BMI of all recruits in 2008.
2. Estimate the percentage of the intake with a BMI greater than 30.0.
3. The BMIs of two randomly chosen recruits are denoted by \(\boldsymbol { B } _ { 1 }\) and \(\boldsymbol { B } _ { 2 }\). Estimate \(\mathrm { P } \left( \boldsymbol { B } _ { 1 } - \boldsymbol { B } _ { 2 } < 5 \right)\).
4. State, giving a reason, for which of the above calculations the normality assumption is unnecessary.

7 In order to improve their mathematics results 10 students attended an intensive Summer School course. Each student took a test at the start of the course and a similar test at the end of the course. The table shows the scores achieved in each test. It is desired to test whether there has been an increase in the population mean score.

1. Explain why a two-sample \(t\)-test would not be appropriate.
2. Stating any necessary assumptions, carry out a suitable \(t\)-test at the \(\frac { 1 } { 2 } \%\) significance level.
3. The Summer School director claims that after taking the course the population mean score increases by more than 5 . Is there sufficient evidence for this claim?