CAIE S1 (Statistics 1) 2018 November

1 A group consists of 5 men and 2 women. Find the number of different ways that the group can stand in a line if the women are not next to each other.

2 A fair 6 -sided die has the numbers \(- 1 , - 1,0,0,1,2\) on its faces. A fair 3 -sided spinner has edges numbered \(- 1,0,1\). The die is thrown and the spinner is spun. The number on the uppermost face of the die and the number on the edge on which the spinner comes to rest are noted. The sum of these two numbers is denoted by \(X\).

1. Draw up a table showing the probability distribution of \(X\).
2. Find \(\operatorname { Var } ( X )\).

3 A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A second ball is now taken at random from the box.

1. Complete the tree diagram to show all the outcomes and the probability for each branch. First ball
   Second ball
   \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_655_392_688_474}
   \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_785_387_703_1110}
2. Find the probability that the two balls taken are the same colour.
3. Find the probability that the first ball taken is red, given that the second ball taken is blue.

4 Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.

1. Find the probability that the group of 7 includes one particular boy.
2. Find the probability that the group of 7 includes at least 2 girls.

5 The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as 'small'; apples weighing more than 140 grams are graded as 'large'; the remainder are graded as 'medium'.

1. Show that the probability that an apple chosen at random is graded as medium is 0.692 , correct to 3 significant figures.
2. Four apples are chosen at random. Find the probability that at least two are graded as medium. [4]

6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .

1. Find the value of \(\sigma\).
   New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
3. Justify the use of your approximate distribution in part (ii).

7 The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
2. Find the median and the interquartile range for the heights of the Anvils.
   The heights of the 11 members of the Anvils are denoted by \(x \mathrm {~cm}\). It is given that \(\Sigma x = 1923\) and \(\Sigma x ^ { 2 } = 337221\). The Anvils are joined by 3 new members whose heights are \(166 \mathrm {~cm} , 172 \mathrm {~cm}\) and 182 cm .
3. Find the standard deviation of the heights of all 14 members of the Anvils.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.