CAIE S1 (Statistics 1) 2003 November

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

2 The floor areas, \(x \mathrm {~m} ^ { 2 }\), of 20 factories are as follows. Represent these data by a histogram on graph paper, using intervals $$0 \leqslant x < 500,500 \leqslant x < 1000,1000 \leqslant x < 2000,2000 \leqslant x < 3000,3000 \leqslant x < 4000 .$$

3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.

4 Single cards, chosen at random, are given away with bars of chocolate. Each card shows a picture of one of 20 different football players. Richard needs just one picture to complete his collection. He buys 5 bars of chocolate and looks at all the pictures. Find the probability that

1. Richard does not complete his collection,
2. he has the required picture exactly once,
3. he completes his collection with the third picture he looks at.

5 In a certain country \(54 \%\) of the population is male. It is known that \(5 \%\) of the males are colour-blind and \(2 \%\) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.

6

1. A collection of 18 books contains one Harry Potter book. Linda is going to choose 6 of these books to take on holiday.
   1. In how many ways can she choose 6 books?
   2. How many of these choices will include the Harry Potter book?
2. In how many ways can 5 boys and 3 girls stand in a straight line
   1. if there are no restrictions,
   2. if the boys stand next to each other?

7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.

1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
   \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).

8 A discrete random variable \(X\) has the following probability distribution.

1. Find the value of the constant \(c\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).