CAIE S1 (Statistics 1) 2014 June

1 Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the data.
2. Make two comparisons between the estimates of the adults and the children.

2 There is a probability of \(\frac { 1 } { 7 }\) that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.

1. Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.
2. Give a reason why the use of a normal approximation is justified.

3 A pet shop has 6 rabbits and 3 hamsters. 5 of these pets are chosen at random. The random variable \(X\) represents the number of hamsters chosen.

1. Show that the probability that exactly 2 hamsters are chosen is \(\frac { 10 } { 21 }\).
2. Draw up the probability distribution table for \(X\).

4 The heights, \(x \mathrm {~cm}\), of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.

1. Find the mean height of the remaining group of 27 people.
2. Find \(\Sigma x ^ { 2 }\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.

5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.

1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.

6 Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3 . Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.

1. Find the probability that Ben becomes the champion after playing exactly 2 games.
2. Find the probability that Ben becomes the champion.
3. Given that Tom becomes the champion, find the probability that he won the 2nd game.

7 Nine cards are numbered \(1,2,2,3,3,4,6,6,6\).

1. All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way
   (a) if the even digits are all together,
   (b) if the first and last digits are both odd.
2. Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way
   (a) if there are no repeated digits,
   (b) if the number is between 200 and 300 .