CAIE S1 (Statistics 1) 2006 June

1 The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be: $$40 , \quad 42 , \quad 45 , \quad 41 , \quad 352 , \quad 40 , \quad 50 , \quad 48 , \quad 51 , \quad 49 , \quad 47 .$$ Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable.

2 The probability that Henk goes swimming on any day is 0.2 . On a day when he goes swimming, the probability that Henk has burgers for supper is 0.75 . On a day when he does not go swimming the probability that he has burgers for supper is \(x\). This information is shown on the following tree diagram.
\includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-2_693_1038_845_555} The probability that Henk has burgers for supper on any day is 0.5 .

1. Find \(x\).
2. Given that Henk has burgers for supper, find the probability that he went swimming that day.

3 The lengths of fish of a certain type have a normal distribution with mean 38 cm . It is found that \(5 \%\) of the fish are longer than 50 cm .

1. Find the standard deviation.
2. When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of fish rejected.
3. 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm .

4
\includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_277_682_274_733} The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.

1. How many possible seating arrangements are there for the 11 passengers?
2. How many possible seating arrangements are there if 5 particular people sit in the back row? Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.
3. In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?

5 Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information.
\includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_789_1627_1468_260}

1. What is the modal age group?
2. How many fathers were between 25 and 30 years old when their first child was born?
3. How many fathers were in the sample?
4. Find the probability that a father, chosen at random from the group, was between 25 and 30 years old when his first child was born, given that he was older than 25 years. 632 teams enter for a knockout competition, in which each match results in one team winning and the other team losing. After each match the winning team goes on to the next round, and the losing team takes no further part in the competition. Thus 16 teams play in the second round, 8 teams play in the third round, and so on, until 2 teams play in the final round.

1. How many teams play in only 1 match?
2. How many teams play in exactly 2 matches?
3. Draw up a frequency table for the numbers of matches which the teams play.
4. Calculate the mean and variance of the numbers of matches which the teams play.

7 A survey of adults in a certain large town found that \(76 \%\) of people wore a watch on their left wrist, \(15 \%\) wore a watch on their right wrist and \(9 \%\) did not wear a watch.

1. A random sample of 14 adults was taken. Find the probability that more than 2 adults did not wear a watch.
2. A random sample of 200 adults was taken. Using a suitable approximation, find the probability that more than 155 wore a watch on their left wrist.