CAIE S1 (Statistics 1) 2015 November

1 For \(n\) values of the variable \(x\), it is given that \(\Sigma ( x - 100 ) = 216\) and \(\Sigma x = 2416\). Find the value of \(n\).

2 A committee of 6 people is to be chosen at random from 7 men and 9 women. Find the probability that there are no men on the committee.

3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that

1. he gets a green robot on opening his first packet,
2. he gets his first green robot on opening his fifth packet. Nick's friend Amos is also collecting robots.
3. Find the probability that the first four packets Amos opens all contain different coloured robots.

4 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]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_284_467_1491_495}
   \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_286_471_1489_1183} 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.

5 The weights, in kilograms, of the 15 rugby players in each of two teams, \(A\) and \(B\), are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with team \(A\) on the lefthand side of the diagram and team \(B\) on the right-hand side.
2. Find the interquartile range of the weights of the players in team \(A\).
3. A new player joins team \(B\) as a substitute. The mean weight of the 16 players in team \(B\) is now 93.9 kg . Find the weight of the new player.

6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.

1. Copy and complete the table showing the possible values of \(X\).

   	Spinner \(A\)
   \cline { 2 - 6 }		1	2	3	3
   Spinner \(B\)	- 2
   - 2			1
   - 1
   1

2. Draw up a table showing the probability distribution of \(X\).
3. Find \(\operatorname { Var } ( X )\).
4. Find the probability that \(X\) is even, given that \(X\) is positive.

7

1. 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.
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).