CAIE S2 (Statistics 2) 2009 June

1 In Europe the diameters of women's rings have mean 18.5 mm . Researchers claim that women in Jakarta have smaller fingers than women in Europe. The researchers took a random sample of 20 women in Jakarta and measured the diameters of their rings. The mean diameter was found to be 18.1 mm . Assuming that the diameters of women's rings in Jakarta have a normal distribution with standard deviation 1.1 mm , carry out a hypothesis test at the \(2 \frac { 1 } { 2 } \%\) level to determine whether the researchers' claim is justified.

2 The weights in grams of oranges grown in a certain area are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). A random sample of 50 of these oranges was taken, and a \(97 \%\) confidence interval for \(\mu\) based on this sample was (222.1, 232.1).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Estimate the sample size that would be required in order for a \(97 \%\) confidence interval for \(\mu\) to have width 8 .

3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.

1. Find the probability that more than 3 major avalanches occur in a 3-month period.
2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.

4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.

1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.

5 The time in minutes taken by candidates to answer a question in an examination has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 6 t - t ^ { 2 } \right) & 3 \leqslant t \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 18 }\).
2. Find the mean time.
3. Find the probability that a candidate, chosen at random, takes longer than 5 minutes to answer the question.
4. Is the upper quartile of the times greater than 5 minutes, equal to 5 minutes or less than 5 minutes? Give a reason for your answer.

6 When Sunil travels from his home in England to visit his relatives in India, his journey is in four stages. The times, in hours, for the stages have independent normal distributions as follows. Bus from home to the airport: \(\quad \mathrm { N } ( 3.75,1.45 )\)
Waiting in the airport: \(\quad \mathrm { N } ( 3.1,0.785 )\)
Flight from England to India: \(\quad \mathrm { N } ( 11,1.3 )\)
Car in India to relatives: \(\quad \mathrm { N } ( 3.2,0.81 )\)

1. Find the probability that the flight time is shorter than the total time for the other three stages.
2. Find the probability that, for 6 journeys to India, the mean time waiting in the airport is less than 4 hours.