CAIE S1 (Statistics 1) 2014 November

1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]

2 A traffic camera measured the speeds, \(x\) kilometres per hour, of 8 cars travelling along a certain street, with the following results. $$\begin{array} { l l l l l l l l } 62.7 & 59.6 & 64.2 & 61.5 & 68.3 & 66.9 & 62.0 & 62.3 \end{array}$$

1. Find \(\Sigma ( x - 62 )\).
2. Find \(\Sigma ( x - 62 ) ^ { 2 }\).
3. Find the mean and variance of the speeds of the 8 cars.

3 The number of books read by members of a book club each year has the binomial distribution \(B ( 12,0.7 )\).

1. State the greatest number of books that could be read by a member of the book club in a particular year and find the probability that a member reads this number of books.
2. Find the probability that a member reads fewer than 10 books in a particular year.

4 A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.

1. Draw a stem-and-leaf diagram to represent the data.
2. On graph paper draw a box-and-whisker plot to represent the data.

5 Gem stones from a certain mine have weights, \(X\) grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g . These gem stones are sorted into three categories for sale depending on their weights, as follows. Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g

1. Find the proportion of gem stones in each of these three categories.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 2.5 ) = 0.8\).

6

1. Seven fair dice each with faces marked 1,2,3,4,5,6 are thrown and placed in a line. Find the number of possible arrangements where the sum of the numbers at each end of the line add up to 4 .
2. Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi and Hebe so that each person receives an odd number of computer games.

7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.