CAIE S1 (Statistics 1) 2015 November

1 The time taken, \(t\) hours, to deliver letters on a particular route each day is measured on 250 working days. The mean time taken is 2.8 hours. Given that \(\Sigma ( t - 2.5 ) ^ { 2 } = 96.1\), find the standard deviation of the times taken.

2 In country \(X , 25 \%\) of people have fair hair. In country \(Y , 60 \%\) of people have fair hair. There are 20 million people in country \(X\) and 8 million people in country \(Y\). A person is chosen at random from these 28 million people.

1. Find the probability that the person chosen is from country \(X\).
2. Find the probability that the person chosen has fair hair.
3. Find the probability that the person chosen is from country \(X\), given that the person has fair hair.

3 Ellie throws two fair tetrahedral dice, each with faces numbered 1, 2, 3 and 4. She notes the numbers on the faces that the dice land on. Event \(S\) is 'the sum of the two numbers is 4 '. Event \(T\) is 'the product of the two numbers is an odd number'.

1. Determine whether events \(S\) and \(T\) are independent, showing your working.
2. Are events \(S\) and \(T\) exclusive? Justify your answer.

4 The time taken for cucumber seeds to germinate under certain conditions has a normal distribution with mean 125 hours and standard deviation \(\sigma\) hours.

1. It is found that \(13 \%\) of seeds take longer than 136 hours to germinate. Find the value of \(\sigma\).
2. 170 seeds are sown. Find the expected number of seeds which take between 131 and 141 hours to germinate.

5

1. Find the number of different ways that the 13 letters of the word ACCOMMODATION can be arranged in a line if all the vowels (A, I, O) are next to each other.
2. There are 7 Chinese, 6 European and 4 American students at an international conference. Four of the students are to be chosen to take part in a television broadcast. Find the number of different ways the students can be chosen if at least one Chinese and at least one European student are included.

6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

1. Draw a histogram on graph paper to illustrate the data.
2. Calculate estimates of the mean and standard deviation of these heights.

7 A factory makes water pistols, \(8 \%\) of which do not work properly.

1. A random sample of 19 water pistols is taken. Find the probability that at most 2 do not work properly.
2. In a random sample of \(n\) water pistols, the probability that at least one does not work properly is greater than 0.9 . Find the smallest possible value of \(n\).
3. A random sample of 1800 water pistols is taken. Use an approximation to find the probability that there are at least 152 that do not work properly.
4. Justify the use of your approximation in part (iii).