CAIE S1 (Statistics 1) 2018 November

2 A random variable \(X\) has the probability distribution shown in the following table, where \(p\) is a constant.

1. Find the value of \(p\).
2. Given that \(\mathrm { E } ( X ) = 1.15\), find \(\operatorname { Var } ( X )\).

3 In an orchestra, there are 11 violinists, 5 cellists and 4 double bass players. A small group of 6 musicians is to be selected from these 20.

1. How many different selections of 6 musicians can be made if there must be at least 4 violinists, at least 1 cellist and no more than 1 double bass player?
   The small group that is selected contains 4 violinists, 1 cellist and 1 double bass player. They sit in a line to perform a concert.
   [0pt]
2. How many different arrangements are there of these 6 musicians if the violinists must sit together? [3]

4

1. It is given that \(X \sim \mathrm {~N} ( 31.4,3.6 )\). Find the probability that a randomly chosen value of \(X\) is less than 29.4.
2. The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than 12 cm long and 58 are more than 19 cm long. Find estimates for the mean and standard deviation of the lengths of fish of this species.

5 At the Nonland Business College, all students sit an accountancy examination at the end of their first year of study. On average, \(80 \%\) of the students pass this examination.

1. A random sample of 9 students who will take this examination is chosen. Find the probability that at most 6 of these students will pass the examination.
2. A random sample of 200 students who will take this examination is chosen. Use a suitable approximate distribution to find the probability that more than 166 of them will pass the examination.
3. Justify the use of your approximate distribution in part (ii).

6 The daily rainfall, \(x \mathrm {~mm}\), in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.

1. On the grid, draw a cumulative frequency graph to illustrate the data.
2. On 100 of the days, the rainfall was \(k \mathrm {~mm}\) or more. Use your graph to estimate the value of \(k\).
3. Calculate estimates of the mean and standard deviation of the daily rainfall in this village.

7 In a group of students, the numbers of boys and girls studying Art, Music and Drama are given in the following table. Each of these 160 students is studying exactly one of these subjects.

1. Find the probability that a randomly chosen student is studying Music.
2. Determine whether the events 'a randomly chosen student is a boy' and 'a randomly chosen student is studying Music' are independent, justifying your answer.
3. Find the probability that a randomly chosen student is not studying Drama, given that the student is a girl.
4. Three students are chosen at random. Find the probability that exactly 1 is studying Music and exactly 2 are boys.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.