CAIE S2 (Statistics 2) 2008 June

1 A magazine conducted a survey about the sleeping time of adults. A random sample of 12 adults was chosen from the adults travelling to work on a train.

1. Give a reason why this is an unsatisfactory sample for the purposes of the survey.
2. State a population for which this sample would be satisfactory. A satisfactory sample of 12 adults gave numbers of hours of sleep as shown below.
   \(4.6 \quad 6.8\)
   5.2
   6.2
   5.7
   \(\begin{array} { l l } 7.1 & 6.3 \end{array}\)
   5.6
   7.0
   \(5.8 \quad 6.5\)
   7.2
3. Calculate unbiased estimates of the mean and variance of the sleeping times of adults.

2 The lengths of time people take to complete a certain type of puzzle are normally distributed with mean 48.8 minutes and standard deviation 15.6 minutes. The random variable \(X\) represents the time taken in minutes by a randomly chosen person to solve this type of puzzle. The times taken by random samples of 5 people are noted. The mean time \(\bar { X }\) is calculated for each sample.

1. State the distribution of \(\bar { X }\), giving the values of any parameters.
2. Find \(\mathrm { P } ( \bar { X } < 50 )\).

3 The lengths of red pencils are normally distributed with mean 6.5 cm and standard deviation 0.23 cm .

1. Two red pencils are chosen at random. Find the probability that their total length is greater than 12.5 cm . The lengths of black pencils are normally distributed with mean 11.3 cm and standard deviation 0.46 cm .
2. Find the probability that the total length of 3 red pencils is more than 6.7 cm greater than the length of 1 black pencil.

4 People who diet can expect to lose an average of 3 kg in a month. In a book, the authors claim that people who follow a new diet will lose an average of more than 3 kg in a month. The weight losses of the 180 people in a random sample who had followed the new diet for a month were noted. The mean was 3.3 kg and the standard deviation was 2.8 kg .

1. Test the authors' claim at the \(5 \%\) significance level, stating your null and alternative hypotheses.
2. State what is meant by a Type II error in words relating to the context of the test in part (i).

5 When a guitar is played regularly, a string breaks on average once every 15 months. Broken strings occur at random times and independently of each other.

1. Show that the mean number of broken strings in a 5 -year period is 4 . A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of 5 years.
2. The mean number of broken strings of the new type in a 5 -year period is denoted by \(\lambda\). Find the rejection region for a test at the \(10 \%\) significance level when the null hypothesis \(\lambda = 4\) is tested against the alternative hypothesis \(\lambda < 4\).
3. Hence calculate the probability of making a Type I error. The number of broken guitar strings of the new type, in a 5 -year period, was in fact 1 .
4. State, with a reason, whether there is evidence at the \(10 \%\) significance level that guitar strings of the new type break less frequently.

6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.

1. Find the probability that exactly two people arrive in a 1-minute period.
2. Find the probability that nobody arrives in a 15 -second period.
3. The probability that at least one person arrives in the next \(t\) minutes is 0.9 . Find the value of \(t\).

7 If Usha is stung by a bee she always develops an allergic reaction. The time taken in minutes for Usha to develop the reaction can be modelled using the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { k } { t + 1 } & 0 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { \ln 5 }\).
2. Find the probability that it takes more than 3 minutes for Usha to develop a reaction.
3. Find the median time for Usha to develop a reaction.