CAIE S1 (Statistics 1) 2002 June

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

1. independent, [2]
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. [6]

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 \(A\). [3]
2. Find \(\text{E}(A)\) and \(\text{Var}(A)\). [4]

1. In a spot check of the speeds \(x \text{ km h}^{-1}\) 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]
2. On another day the mean speed of cars on the motorway was found to be \(107.6 \text{ km h}^{-1}\) and the standard deviation was \(13.8 \text{ km h}^{-1}\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \text{ km h}^{-1}\), find what proportion of cars exceed the speed limit. [3]

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]
2. the digits 1, 3, 7 (in any order) are next to each other, [3]
3. these 7-digit numbers are even. [3]

1. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), \(\text{P}(X > 3.6) = 0.5\) and \(\text{P}(X > 2.8) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\). [4]
2. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]

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.
   1. Find the number of plants per box. [4]
   2. Find the probability that a box contains exactly 12 plants which produce yellow flowers. [2]
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. [4]