CAIE S1 (Statistics 1) 2011 November

1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.

2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .

1. Find the standard deviation of \(x\).
2. Find \(\Sigma x ^ { 2 }\).

3 A team of 4 is to be randomly chosen from 3 boys and 5 girls. The random variable \(X\) is the number of girls in the team.

1. Draw up a probability distribution table for \(X\).
2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 2 }\), calculate \(\operatorname { Var } ( X )\).

4 The marks of the pupils in a certain class in a History examination are as follows. $$\begin{array} { l l l l l l l l l l l l l } 28 & 33 & 55 & 38 & 42 & 39 & 27 & 48 & 51 & 37 & 57 & 49 & 33 \end{array}$$ The marks of the pupils in a Physics examination are summarised as follows.
Lower quartile: 28 , Median: 39, Upper quartile: 67.
The lowest mark was 17 and the highest mark was 74 .

1. Draw box-and-whisker plots in a single diagram on graph paper to illustrate the marks for History and Physics.
2. State one difference, which can be seen from the diagram, between the marks for History and Physics.

5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.

1. Find the standard deviation of the weights of the letters.
2. Find the probability that a randomly chosen letter weighs more than 13 g .
3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.

6

1. Find the number of different ways in which the 12 letters of the word STRAWBERRIES can be arranged
   1. if there are no restrictions,
   2. if the 4 vowels \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { I }\) must all be together.
2. 1. 4 astronauts are chosen from a certain number of candidates. If order of choosing is not taken into account, the number of ways the astronauts can be chosen is 3876 . How many ways are there if order of choosing is taken into account?
   2. 4 astronauts are chosen to go on a mission. Each of these astronauts can take 3 personal possessions with him. How many different ways can these 12 possessions be arranged in a row if each astronaut's possessions are kept together?

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.

1. Find the probability that exactly two of the selected balls have the same number.
2. Given that exactly two of the selected balls have the same number, find the probability that they are both numbered 2 .
3. Event \(X\) is 'exactly two of the selected balls have the same number'. Event \(Y\) is 'the ball selected from bag \(A\) has number 2'. Showing your working, determine whether events \(X\) and \(Y\) are independent or not.