CAIE S2 (Statistics 2) 2003 November

1 The result of a memory test is known to be normally distributed with mean \(\mu\) and standard deviation 1.9. It is required to have a \(95 \%\) confidence interval for \(\mu\) with a total width of less than 2.0 . Find the least possible number of tests needed to achieve this.

2 A certain machine makes matches. One match in 10000 on average is defective. Using a suitable approximation, calculate the probability that a random sample of 45000 matches will include 2,3 or 4 defective matches.

3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.

4 The number of emergency telephone calls to the electricity board office in a certain area in \(t\) minutes is known to follow a Poisson distribution with mean \(\frac { 1 } { 80 } t\).

1. Find the probability that there will be at least 3 emergency telephone calls to the office in any 20-minute period.
2. The probability that no emergency telephone call is made to the office in a period of \(k\) minutes is 0.9 . Find \(k\).

5 The distance driven in a week by a long-distance lorry driver is a normally distributed random variable with mean 1850 km and standard deviation 117 km .

1. Find the probability that in a random sample of 26 weeks his average distance driven per week is more than 1800 km .
2. New driving regulations are introduced and in a random sample of 26 weeks after their introduction the lorry driver drives a total of 47658 km . Assuming the standard deviation remains unchanged, test at the \(10 \%\) level whether his mean weekly driving distance has changed.

6

1. Explain what is meant by
   (a) a Type I error,
   (b) a Type II error.
2. Roger thinks that a box contains 6 screws and 94 nails. Felix thinks that the box contains 30 screws and 70 nails. In order to test these assumptions they decide to take 5 items at random from the box and inspect them, replacing each item after it has been inspected, and accept Roger's hypothesis (the null hypothesis) if all 5 items are nails.
   (a) Calculate the probability of a Type I error.
   (b) If Felix's hypothesis (the alternative hypothesis) is true, calculate the probability of a Type II error.

7 The lifetime, \(x\) years, of the power light on a freezer, which is left on continuously, can be modelled by the continuous random variable with density function given by $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - 3 x } & x > 0
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 3\).
2. Find the lower quartile.
3. Find the mean lifetime.