CAIE S1 (Statistics 1) 2007 June

1 The length of time, \(t\) minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. $$\Sigma ( t - 35 ) = - 15 \quad \Sigma ( t - 35 ) ^ { 2 } = 82.23$$ Calculate the mean and standard deviation of these times taken to do the crossword.

2 Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team.

1. Find the probability that Jamie is chosen for the team.
2. Find the conditional probability that Jamie attended the training session, given that he was chosen for the team.

3

1. The random variable \(X\) is normally distributed. The mean is twice the standard deviation. It is given that \(\mathrm { P } ( X > 5.2 ) = 0.9\). Find the standard deviation.
2. A normal distribution has mean \(\mu\) and standard deviation \(\sigma\). If 800 observations are taken from this distribution, how many would you expect to be between \(\mu - \sigma\) and \(\mu + \sigma\) ?

4 The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9 -year-olds and by the members of a class of eight 16 -year-olds are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the information above.
2. A new pupil joined the 16 -year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil's time to swim the distance.

5

1. Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
   (a) if there are no restrictions,
   (b) if the Rs must all be together.
2. How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?

6 The probability that New Year's Day is on a Saturday in a randomly chosen year is \(\frac { 1 } { 7 }\).

1. 15 years are chosen randomly. Find the probability that at least 3 of these years have New Year's Day on a Saturday.
2. 56 years are chosen randomly. Use a suitable approximation to find the probability that more than 7 of these years have New Year's Day on a Saturday.

7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket.

1. Find the probability that the three peppers are all different colours.
2. Show that the probability that exactly 2 of the peppers taken are green is \(\frac { 12 } { 55 }\).
3. The number of green peppers taken is denoted by the discrete random variable \(X\). Draw up a probability distribution table for \(X\).