CAIE S1 (Statistics 1) 2016 June

1 The height of maize plants in Mpapwa is normally distributed with mean 1.62 m and standard deviation \(\sigma \mathrm { m }\). The probability that a randomly chosen plant has a height greater than 1.8 m is 0.15 . Find the value of \(\sigma\).

2 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The random variable \(X\) is the score when the die is thrown. The following is the probability distribution table for \(X\). The die is thrown 3 times. Find the probability that the score is 4 on not more than 1 of the 3 throws.

3 The probability that the school bus is on time on any particular day is 0.6 . If the bus is on time the probability that Sam the driver gets a cup of coffee is 0.9 . If the bus is not on time the probability that Sam gets a cup of coffee is 0.3 .

1. Find the probability that Sam gets a cup of coffee.
2. Given that Sam does not get a cup of coffee, find the probability that the bus is not on time.

4 A box contains 2 green sweets and 5 blue sweets. Two sweets are taken at random from the box, without replacement. The random variable \(X\) is the number of green sweets taken. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm . The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm .

1. A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
2. 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.
3. Justify the use of your approximation.

6

1. 1. Find how many numbers there are between 100 and 999 in which all three digits are different.
   2. Find how many of the numbers in part (i) are odd numbers greater than 700 .
2. A bunch of flowers consists of a mixture of roses, tulips and daffodils. Tom orders a bunch of 7 flowers from a shop to give to a friend. There must be at least 2 of each type of flower. The shop has 6 roses, 5 tulips and 4 daffodils, all different from each other. Find the number of different bunches of flowers that are possible.

7 The amounts spent by 160 shoppers at a supermarket are summarised in the following table.

1. Draw a cumulative frequency graph of this distribution.
2. Estimate the median and the interquartile range of the amount spent.
3. Estimate the number of shoppers who spent more than \(
   ) 115$.
4. Calculate an estimate of the mean amount spent.