CAIE S1 (Statistics 1) 2004 November

1 The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.

1. Find the number of different arrangements using all nine letters.
2. How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?

2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.

1. Find the value of \(a\).
2. Draw a histogram on graph paper to represent the data.
3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.

3 When Andrea needs a taxi, she rings one of three taxi companies, A, B or C. 50\% of her calls are to taxi company \(A , 30 \%\) to \(B\) and \(20 \%\) to \(C\). A taxi from company \(A\) arrives late \(4 \%\) of the time, a taxi from company \(B\) arrives late \(6 \%\) of the time and a taxi from company \(C\) arrives late \(17 \%\) of the time.

1. Find the probability that, when Andrea rings for a taxi, it arrives late.
2. Given that Andrea's taxi arrives late, find the conditional probability that she rang company \(B\).

4 The ages, \(x\) years, of 18 people attending an evening class are summarised by the following totals: \(\Sigma x = 745 , \Sigma x ^ { 2 } = 33951\).

1. Calculate the mean and standard deviation of the ages of this group of people.
2. One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find the age of the person who left and the standard deviation of the ages of the remaining 17 people.

5 The length of Paulo's lunch break follows a normal distribution with mean \(\mu\) minutes and standard deviation 5 minutes. On one day in four, on average, his lunch break lasts for more than 52 minutes.

1. Find the value of \(\mu\).
2. Find the probability that Paulo's lunch break lasts for between 40 and 46 minutes on every one of the next four days.

6 A box contains five balls numbered \(1,2,3,4,5\). Three balls are drawn randomly at the same time from the box.

1. By listing all possible outcomes (123, 124, etc.), find the probability that the sum of the three numbers drawn is an odd number. The random variable \(L\) denotes the largest of the three numbers drawn.
2. Find the probability that \(L\) is 4 .
3. Draw up a table to show the probability distribution of \(L\).
4. Calculate the expectation and variance of \(L\).

7

1. State two conditions which must be satisfied for a situation to be modelled by a binomial distribution. In a certain village 28\% of all cars are made by Ford.
2. 14 cars are chosen randomly in this village. Find the probability that fewer than 4 of these cars are made by Ford.
3. A random sample of 50 cars in the village is taken. Estimate, using a normal approximation, the probability that more than 18 cars are made by Ford.