CAIE S2 (Statistics 2) 2005 November

1 The number of words on a page of a book can be modelled by a normal distribution with mean 403 and standard deviation 26.8. Find the probability that the average number of words per page in a random sample of 6 pages is less than 410.

2 A manufacturer claims that \(20 \%\) of sugar-coated chocolate beans are red. George suspects that this percentage is actually less than \(20 \%\) and so he takes a random sample of 15 chocolate beans and performs a hypothesis test with the null hypothesis \(p = 0.2\) against the alternative hypothesis \(p < 0.2\). He decides to reject the null hypothesis in favour of the alternative hypothesis if there are 0 or 1 red beans in the sample.

1. With reference to this situation, explain what is meant by a Type I error.
2. Find the probability of a Type I error in George's test.

3 Flies stick to wet paint at random points. The average number of flies is 2 per square metre. A wall with area \(22 \mathrm {~m} ^ { 2 }\) is painted with a new type of paint which the manufacturer claims is fly-repellent. It is found that 27 flies stick to this wall. Use a suitable approximation to test the manufacturer's claim at the \(1 \%\) significance level. Take the null hypothesis to be \(\mu = 44\), where \(\mu\) is the population mean.

4

1. Give a reason why, in carrying out a statistical investigation, a sample rather than a complete population may be used.
2. Rose wishes to investigate whether men in her town have a different life-span from the national average of 71.2 years. She looks at government records for her town and takes a random sample of the ages of 110 men who have died recently. Their mean age in years was 69.3 and the unbiased estimate of the population variance was 65.61.
   (a) Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
   (b) State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.

5 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 3 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that the value of \(a\) is \(\frac { 1 } { 2 }\).
2. Find \(\mathrm { P } ( X > 1.8 )\).
3. Find \(\mathrm { E } ( X )\).

6 A shopkeeper sells electric fans. The demand for fans follows a Poisson distribution with mean 3.2 per week.

1. Find the probability that the demand is exactly 2 fans in any one week.
2. The shopkeeper has 4 fans in his shop at the beginning of a week. Find the probability that this will not be enough to satisfy the demand for fans in that week.
3. Given instead that he has \(n\) fans in his shop at the beginning of a week, find, by trial and error, the least value of \(n\) for which the probability of his not being able to satisfy the demand for fans in that week is less than 0.05 .

7 A journey in a certain car consists of two stages with a stop for filling up with fuel after the first stage. The length of time, \(T\) minutes, taken for each stage has a normal distribution with mean 74 and standard deviation 7.3. The length of time, \(F\) minutes, it takes to fill up with fuel has a normal distribution with mean 5 and standard deviation 1.7. The length of time it takes to pay for the fuel is exactly 4 minutes. The variables \(T\) and \(F\) are independent and the times for the two stages are independent of each other.

1. Find the probability that the total time for the journey is less than 154 minutes.
2. A second car has a fuel tank with exactly twice the capacity of the first car. Find the mean and variance of this car's fuel fill-up time.
3. This second car's time for each stage of the journey follows a normal distribution with mean 69 minutes and standard deviation 5.2 minutes. The length of time it takes to pay for the fuel for this car is also exactly 4 minutes. Find the probability that the total time for the journey taken by the first car is more than the total time taken by the second car.