CAIE S1 (Statistics 1) 2013 June

1 The random variable \(Y\) is normally distributed with mean equal to five times the standard deviation. It is given that \(\mathrm { P } ( Y > 20 ) = 0.0732\). Find the mean.

2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \quad \text { and } \quad \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).

3 Cans of lemon juice are supposed to contain 440 ml of juice. It is found that the actual volume of juice in a can is normally distributed with mean 445 ml and standard deviation 3.6 ml .

1. Find the probability that a randomly chosen can contains less than 440 ml of juice. It is found that \(94 \%\) of the cans contain between \(( 445 - c ) \mathrm { ml }\) and \(( 445 + c ) \mathrm { ml }\) of juice.
2. Find the value of \(c\).

4 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.

1. Find the probability that at least 2 of the 5 integers are less than or equal to 4 . Robert now generates \(n\) random integers between 1 and 9 inclusive. The random variable \(X\) is the number of these \(n\) integers which are less than or equal to a certain integer \(k\) between 1 and 9 inclusive. It is given that the mean of \(X\) is 96 and the variance of \(X\) is 32 .
2. Find the values of \(n\) and \(k\).

5 The following are the annual amounts of money spent on clothes, to the nearest \(
) 10$, by 27 people.

1. Construct a stem-and-leaf diagram for the data.
2. Find the median and the interquartile range of the data. An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
3. List the outliers.

6 A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.

1. How many different selections of 12 trees can be made if there must be at least 2 of each type of tree? The council buys 4 hibiscus trees, 6 jacaranda trees and 2 oleander trees.
2. How many different arrangements of these 12 trees can be made if the hibiscus trees have to be next to each other, the jacaranda trees have to be next to each other and the oleander trees have to be next to each other?
3. How many different arrangements of these 12 trees can be made if no hibiscus tree is next to another hibiscus tree?

7 Susan has a bag of sweets containing 7 chocolates and 5 toffees. Ahmad has a bag of sweets containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Susan's bag and put in Ahmad's bag. A sweet is then taken at random from Ahmad's bag.

1. Find the probability that the two sweets taken are a toffee from Susan's bag and a boiled sweet from Ahmad's bag.
2. Given that the sweet taken from Ahmad's bag is a chocolate, find the probability that the sweet taken from Susan's bag was also a chocolate.
3. The random variable \(X\) is the number of times a chocolate is taken. State the possible values of \(X\) and draw up a table to show the probability distribution of \(X\).