CAIE S1 (Statistics 1) 2019 November

1 Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows. $$\begin{array} { l l l l l l l l l l l l } 50 & 45 & 62 & 30 & 40 & 55 & 110 & 38 & 52 & 60 & 55 & 40 \end{array}$$

1. Find the median and the interquartile range for the data.
2. Give a disadvantage of using the mean as a measure of the central tendency in this case.

2 Benju cycles to work each morning and he has two possible routes. He chooses the hilly route with probability 0.4 and the busy route with probability 0.6 . If he chooses the hilly route, the probability that he will be late for work is \(x\) and if he chooses the busy route the probability that he will be late for work is \(2 x\). The probability that Benju is late for work on any day is 0.36 .

1. Show that \(x = 0.225\).
2. Given that Benju is not late for work, find the probability that he chooses the hilly route.

3 The speeds, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results are summarised in the following table.

1. On the grid, draw a histogram to illustrate the data in the table.
   \includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker.

4 In Quarendon, \(66 \%\) of households are satisfied with the speed of their wifi connection.

1. Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
2. A random sample of 150 households in Quarendon is chosen. Use a suitable approximation to find the probability that more than 84 are satisfied with the speed of their wifi connection. [5]

5 A fair red spinner has four sides, numbered 1, 2, 3, 3. A fair blue spinner has three sides, numbered \(- 1,0,2\). When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).

6 The heights, in metres, of fir trees in a large forest have a normal distribution with mean 40 and standard deviation 8 .

1. Find the probability that a fir tree chosen at random in this forest has a height less than 45 metres.
2. Find the probability that a fir tree chosen at random in this forest has a height within 5 metres of the mean.
   In another forest, the heights of another type of fir tree are modelled by a normal distribution. A scientist measures the heights of 500 randomly chosen trees of this type. He finds that 48 trees are less than 10 m high and 76 trees are more than 24 m high.
3. Find the mean and standard deviation of the heights of trees of this type.

7

1. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.
2. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.
3. Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.
4. Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.