CAIE S1 (Statistics 1) 2018 November

1

1. How many different arrangements are there of the 11 letters in the word MISSISSIPPI?
2. Two letters are chosen at random from the 11 letters in the word MISSISSIPPI. Find the probability that these two letters are the same.

2 The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, \(A\) and \(B\). Key: 5 | 22 | 6 means a reaction time of 0.225 seconds for \(A\) and 0.226 seconds for \(B\)

1. Find the median and the interquartile range for group \(A\).
   The median value for group \(B\) is 0.235 seconds, the lower quartile is 0.217 seconds and the upper quartile is 0.245 seconds.
2. Draw box-and-whisker plots for groups \(A\) and \(B\) on the grid.
   \includegraphics[max width=\textwidth, alt={}, center]{62812433-baee-490a-bad4-b6b0f917c234-03_805_1495_1729_365}

3 Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75 , independently of all other days.

1. Find the probability that he will complete the puzzle at least three times over a period of five days.
   Kenny also attempts the puzzle every day. The probability that he will complete the puzzle on a Monday is 0.8 . The probability that he will complete it on a Tuesday is 0.9 if he completed it on the previous day and 0.6 if he did not complete it on the previous day.
2. Find the probability that Kenny will complete the puzzle on at least one of the two days Monday and Tuesday in a randomly chosen week.

4

1. Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together.
2. Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy.

5 The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y ^ { 2 } = 42850\), where \(y\) is the age of a Senior member in years.

1. Find the mean age of all 32 members of the club.
2. Find the standard deviation of the ages of all 32 members of the club.

6 A fair red spinner has 4 sides, numbered 1,2,3,4. A fair blue spinner has 3 sides, numbered 1,2,3. When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).
3. Find the probability that \(X\) is equal to 1 , given that \(X\) is non-zero.

7

1. The time, \(X\) hours, for which students use a games machine in any given day has a normal distribution with mean 3.24 hours and standard deviation 0.96 hours.
   1. On how many days of the year ( 365 days) would you expect a randomly chosen student to use a games machine for less than 4 hours?
   2. Find the value of \(k\) such that \(\mathrm { P } ( X > k ) = 0.2\).
   3. Find the probability that the number of hours for which a randomly chosen student uses a games machine in a day is within 1.5 standard deviations of the mean.
2. The variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), where \(4 \sigma = 3 \mu\) and \(\mu \neq 0\). Find the probability that a randomly chosen value of \(Y\) is positive.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.