CAIE S1 (Statistics 1) 2017 November

1 A statistics student asks people to complete a survey. The probability that a randomly chosen person agrees to complete the survey is 0.2 . Find the probability that at least one of the first three people asked agrees to complete the survey.

2 Tien measured the arm lengths, \(x \mathrm {~cm}\), of 20 people in his class. He found that \(\Sigma x = 1218\) and the standard deviation of \(x\) was 4.2. Calculate \(\Sigma ( x - 45 )\) and \(\Sigma ( x - 45 ) ^ { 2 }\).

3 At the end of a revision course in mathematics, students have to pass a test to gain a certificate. The probability of any student passing the test at the first attempt is 0.85 . Those students who fail are allowed to retake the test once, and the probability of any student passing the retake test is 0.65 .

1. Draw a fully labelled tree diagram to show all the outcomes.
2. Given that a student gains the certificate, find the probability that this student fails the test on the first attempt.

4 A fair die with faces numbered \(1,2,2,2,3,6\) is thrown. The score, \(X\), is found by squaring the number on the face the die shows and then subtracting 4.

1. Draw up a table to show the probability distribution of \(X\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 The number of Olympic medals won in the 2012 Olympic Games by the top 27 countries is shown below.

1. Draw a stem-and-leaf diagram to illustrate the data.
2. Find the median and quartiles and draw a box-and-whisker plot on the grid.
   \includegraphics[max width=\textwidth, alt={}, center]{4c2afa86-960c-473e-970c-ed16c8434fec-07_1006_1406_1007_411}

6 A car park has spaces for 18 cars, arranged in a line. On one day there are 5 cars, of different makes, parked in randomly chosen positions and 13 empty spaces.

1. Find the number of possible arrangements of the 5 cars in the car park.
2. Find the probability that the 5 cars are not all next to each other.
   On another day, 12 cars of different makes are parked in the car park. 5 of these cars are red, 4 are white and 3 are black. Elizabeth selects 3 of these cars.
   [0pt]
3. Find the number of selections Elizabeth can make that include cars of at least 2 different colours. [5]

7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).

1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
   On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
2. Find the value of \(x\).
3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
4. Find the probability that Josie misses the bus.