CAIE S1 (Statistics 1) 2002 June

1 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.8\) and \(\mathrm { P } ( A\) and \(B\) )=0.4. State, giving a reason in each case, whether events \(A\) and \(B\) are

1. independent,
2. mutually exclusive.

2 The manager of a company noted the times spent in 80 meetings. The results were as follows. Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range.

3 A fair cubical die with faces numbered \(1,1,1,2,3,4\) is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3 , the value of \(A\) is 9 .

1. Draw up a table to show the probability distribution of \(A\).
2. Find \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
3. In a spot check of the speeds \(x \mathrm {~km} \mathrm {~h} ^ { - 1 }\) of 30 cars on a motorway, the data were summarised by \(\Sigma ( x - 110 ) = - 47.2\) and \(\Sigma ( x - 110 ) ^ { 2 } = 5460\). Calculate the mean and standard deviation of these speeds.
4. On another day the mean speed of cars on the motorway was found to be \(107.6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation was \(13.8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), find what proportion of cars exceed the speed limit.

5 The digits of the number 1223678 can be rearranged to give many different 7 -digit numbers. Find how many different 7-digit numbers can be made if

1. there are no restrictions on the order of the digits,
2. the digits 1,3,7 (in any order) are next to each other,
3. these 7 -digit numbers are even.
4. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma , \mathrm { P } ( X > 3.6 ) = 0.5\) and \(\mathrm { P } ( X > 2.8 ) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\).
5. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8.

7

1. A garden shop sells polyanthus plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95.
   (a) Find the number of plants per box.
   (b) Find the probability that a box contains exactly 12 plants which produce yellow flowers.
2. Another garden shop sells polyanthus plants in boxes of 100 . The shop's advertisement states that the probability of any polyanthus plant producing a pink flower is 0.3 . Use a suitable approximation to find the probability that a box contains fewer than 35 plants which produce pink flowers.