CAIE S2 (Statistics 2) 2010 June

1
\includegraphics[max width=\textwidth, alt={}, center]{29ab8740-9fad-4596-b34f-c424b5464858-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.

1. State the value of \(k\).
2. Explain briefly what this graph tells you about the arrival times of trains.

2 A random sample of \(n\) people were questioned about their internet use. 87 of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is \(0.1129 < p < 0.1771\).

1. Write down the mid-point of this confidence interval and hence find the value of \(n\).
2. This interval is an \(\alpha \%\) confidence interval. Find \(\alpha\).

3 Metal bolts are produced in large numbers and have lengths which are normally distributed with mean 2.62 cm and standard deviation 0.30 cm .

1. Find the probability that a random sample of 45 bolts will have a mean length of more than 2.55 cm .
2. The machine making these bolts is given an annual service. This may change the mean length of bolts produced but does not change the standard deviation. To test whether the mean has changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length is \(m \mathrm {~cm}\). Find the set of values of \(m\) which result in rejection at the \(10 \%\) significance level of the hypothesis that no change in the mean length has occurred.

4 The weekly distance in kilometres driven by Mr Parry has a normal distribution with mean 512 and standard deviation 62. Independently, the weekly distance in kilometres driven by Mrs Parry has a normal distribution with mean 89 and standard deviation 7.4.

1. Find the probability that, in a randomly chosen week, Mr Parry drives more than 5 times as far as Mrs Parry.
2. Find the mean and standard deviation of the total of the weekly distances in miles driven by Mr Parry and Mrs Parry. Use the approximation 8 kilometres \(= 5\) miles.

5 The random variable \(T\) denotes the time in seconds for which a firework burns before exploding. The probability density function of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} k \mathrm { e } ^ { 0.2 t } & 0 \leqslant t \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 5 ( \mathrm { e } - 1 ) }\).
2. Sketch the probability density function.
3. \(80 \%\) of fireworks burn for longer than a certain time before they explode. Find this time.

6 In restaurant \(A\) an average of 2.2\% of tablecloths are stained and, independently, in restaurant \(B\) an average of 5.8\% of tablecloths are stained.

1. Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson approximation to find the probability that a total of more than 2 tablecloths are stained.
2. Random samples of \(n\) tablecloths are taken from each restaurant. The probability that at least one tablecloth is stained is greater than 0.99 . Find the least possible value of \(n\).

7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the \(10 \%\) significance level whether the mean white blood cell count has decreased.

1. State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
2. Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.
3. Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.