CAIE S2 (Statistics 2) 2007 June

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 3.2,1.2 ^ { 2 } \right)\). The sum of 60 independent observations of \(X\) is denoted by \(S\). Find \(\mathrm { P } ( S > 200 )\).

3 A machine has produced nails over a long period of time, where the length in millimetres was distributed as \(\mathrm { N } ( 22.0,0.19 )\). It is believed that recently the mean length has changed. To test this belief a random sample of 8 nails is taken and the mean length is found to be 21.7 mm . Carry out a hypothesis test at the \(5 \%\) significance level to test whether the population mean has changed, assuming that the variance remains the same.

4 At a certain airport 20\% of people take longer than an hour to check in. A new computer system is installed, and it is claimed that this will reduce the time to check in. It is decided to accept the claim if, from a random sample of 22 people, the number taking longer than an hour to check in is either 0 or 1 .

1. Calculate the significance level of the test.
2. State the probability that a Type I error occurs.
3. Calculate the probability that a Type II error occurs if the probability that a person takes longer than an hour to check in is now 0.09 .

5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.

1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
2. Find the probability that exactly 8 people call in any half hour.
3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.

6 The daily takings, \(
) x\(, for a shop were noted on 30 randomly chosen days. The takings are summarised by \)\Sigma x = 31500 , \Sigma x ^ { 2 } = 33141816$.

1. Calculate unbiased estimates of the population mean and variance of the shop's daily takings.
2. Calculate a \(98 \%\) confidence interval for the mean daily takings. The mean daily takings for a random sample of \(n\) days is found.
3. Estimate the value of \(n\) for which it is approximately \(95 \%\) certain that the sample mean does not differ from the population mean by more than \(
   ) 6$.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } \left( x ^ { 2 } - 1 \right) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

1. Sketch the probability density function of \(X\).
2. Show that the mean, \(\mu\), of \(X\) is 1.6875 .
3. Show that the standard deviation, \(\sigma\), of \(X\) is 0.2288 , correct to 4 decimal places.
4. Find \(\mathrm { P } ( 1 \leqslant X \leqslant \mu + \sigma )\).