CAIE S1 (Statistics 1) 2010 June

1 A bottle of sweets contains 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets. 7 sweets are selected at random. Find the probability that exactly 3 of them are red.

2 The heights, \(x \mathrm {~cm}\), of a group of 82 children are summarised as follows. $$\Sigma ( x - 130 ) = - 287 , \quad \text { standard deviation of } x = 6.9 .$$

1. Find the mean height.
2. Find \(\Sigma ( x - 130 ) ^ { 2 }\).

3 Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6 . If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park there is a probability of 0.75 that the dog will bark.

1. Find the probability that they go to the park on more than 5 of the next 7 days.
2. Find the probability that the dog barks on any particular day.
3. Find the variance of the number of times they go to the park in 30 days.

4 Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if

1. the first and last cans in the row are the same type of drink,
2. the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.

5 Set \(A\) consists of the ten digits \(0,0,0,0,0,0,2,2,2,4\).
Set \(B\) consists of the seven digits \(0,0,0,0,2,2,2\).
One digit is chosen at random from each set. The random variable \(X\) is defined as the sum of these two digits.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 3 } { 7 }\).
2. Tabulate the probability distribution of \(X\).
3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
4. Given that \(X = 2\), find the probability that the digit chosen from set \(A\) was 2 .

6 The lengths of some insects of the same type from two countries, \(X\) and \(Y\), were measured. The stem-and-leaf diagram shows the results. Key: 5 | 81 | 3 means an insect from country \(X\) has length 0.815 cm and an insect from country \(Y\) has length 0.813 cm .

1. Find the median and interquartile range of the lengths of the insects from country \(X\).
2. The interquartile range of the lengths of the insects from country \(Y\) is 0.028 cm . Find the values of \(q\) and \(r\).
3. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
4. Compare the lengths of the insects from the two countries.

7 The heights that children of a particular age can jump have a normal distribution. On average, 8 children out of 10 can jump a height of more than 127 cm , and 1 child out of 3 can jump a height of more than 135 cm .

1. Find the mean and standard deviation of the heights the children can jump.
2. Find the probability that a randomly chosen child will not be able to jump a height of 145 cm .
3. Find the probability that, of 8 randomly chosen children, at least 2 will be able to jump a height of more than 135 cm .