CAIE S1 (Statistics 1) 2020 Specimen

1 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males. Key: 2 | 20 | 3 means \\(20200for females and \\)20300 for males.

1. Find the median and the quartiles of the females' salaries.
   You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
2. Draw a pair of box-and-whisker plots in a single diagram on the grid below to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-02_994_1589_1736_310}

2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \text { and } \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).

3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.

1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
2. Draw up the probability distribution table for \(X\).
3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).

4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.

1. Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
   The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
2. Find the value of \(m\).
3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.

5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.

1. Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
2. Justify your use of the approximation in part (a).
   On another occasion, the same die is thrown repeatedly until a 3 is obtained.
3. Find the probability that obtaining a 3 requires fewer than 7 throws.

6 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.

1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_272_456_242_461} \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_281_455_233_1151} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
2. Find the number of different seating arrangements that are now possible for the 8 friends.

7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.

• Event \(X\) is 'exactly two of the selected balls have the same number'.
• Event \(Y\) is 'the ball selected from bag \(A\) has number 4'.
  1. Find \(\mathrm { P } ( X )\).
  2. Find \(\mathrm { P } ( X \cap Y )\) and hence determine whether or not events \(X\) and \(Y\) are independent.
  3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.