CAIE S1 (Statistics 1) 2009 June

1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.

1. Calculate the value of \(\mu\).
2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.

2 Gohan throws a fair tetrahedral die with faces numbered \(1,2,3,4\). If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable \(X\) denote Gohan's score.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 5 } { 16 }\).
2. The table below shows the probability distribution of \(X\).

   \(x\)	2	3	4	5	6	7
   \(\mathrm { P } ( X = x )\)	\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 1 } { 16 }\)

   Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.

1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.

4 A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.

1. In how many different ways can the group be chosen?
2. In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
3. The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?

5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.

1. Find the probabilities that he goes on each of the three rides. The probabilities that Ravi is frightened on each of the rides are as follows: $$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
2. Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened. Ravi goes on a ride.
3. Find the probability that he is frightened.
4. Given that Ravi is not frightened, find the probability that he went on the camel ride.

6 During January the numbers of people entering a store during the first hour after opening were as follows.

1. Find the values of \(a\) and \(b\).
2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
3. Use your graph to estimate the median time after opening that people entered the store.
4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.