CAIE S1 (Statistics 1) 2010 June

1 The probability distribution of the discrete random variable \(X\) is shown in the table below. Given that \(\mathrm { E } ( X ) = 0.75\), find the values of \(a\) and \(b\).

2 The numbers of people travelling on a certain bus at different times of the day are as follows.

1. Draw a stem-and-leaf diagram to illustrate the information given above.
2. Find the median, the lower quartile, the upper quartile and the interquartile range.
3. State, in this case, which of the median and mode is preferable as a measure of central tendency, and why.

3 The random variable \(X\) is the length of time in minutes that Jannon takes to mend a bicycle puncture. \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is given that \(\mathrm { P } ( X > 30.0 ) = 0.1480\) and \(\mathrm { P } ( X > 20.9 ) = 0.6228\). Find \(\mu\) and \(\sigma\).

4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.

1. The mean cost of Fei's rides is \(
   ) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(
   ) 2.50\( per ride. [2]
2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.

5 In the holidays Martin spends \(25 \%\) of the day playing computer games. Martin's friend phones him once a day at a randomly chosen time.

1. Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which Martin is playing computer games when his friend phones.
2. Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a normal approximation to find the probability that there are fewer than 7 days on which Martin is playing computer games when his friend phones.
3. Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is playing computer games when his friend phones.

6

1. Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs.
2. Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other.
3. Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together.

7 In a television quiz show Peter answers questions one after another, stopping as soon as a question is answered wrongly.

• The probability that Peter gives the correct answer himself to any question is 0.7 .
• The probability that Peter gives a wrong answer himself to any question is 0.1 .
• The probability that Peter decides to ask for help for any question is 0.2 .

On the first occasion that Peter decides to ask for help he asks the audience. The probability that the audience gives the correct answer to any question is 0.95 . This information is shown in the tree diagram below.
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\includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_270_743_2010_1023}

1. Show that the probability that the first question is answered correctly is 0.89 . On the second occasion that Peter decides to ask for help he phones a friend. The probability that his friend gives the correct answer to any question is 0.65 .
2. Find the probability that the first two questions are both answered correctly.
3. Given that the first two questions were both answered correctly, find the probability that Peter asked the audience.