CAIE S1 (Statistics 1) 2012 November

1 In a normal distribution with mean 9.3, the probability of a randomly chosen value being greater than 5.6 is 0.85 . Find the standard deviation.

2 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).

3 Ronnie obtained data about the gross domestic product (GDP) and the birth rate for 170 countries. He classified each GDP and each birth rate as either 'low', 'medium' or 'high'. The table shows the number of countries in each category. One of these countries is chosen at random.

1. Find the probability that the country chosen has a medium GDP.
2. Find the probability that the country chosen has a low birth rate, given that it does not have a medium GDP.
3. State with a reason whether or not the events 'the country chosen has a high GDP' and 'the country chosen has a high birth rate' are exclusive. One country is chosen at random from those countries which have a medium GDP and then a different country is chosen at random from those which have a medium birth rate.
4. Find the probability that both countries chosen have a medium GDP and a medium birth rate.

4 In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.

1. Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.
2. Draw, on graph paper, a histogram to illustrate the information in the table.

5 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 82,126 )\).

1. A value of \(X\) is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .
2. Five independent observations of \(X\) are taken. Find the probability that at most one of them is greater than 87.
3. Find the value of \(k\) such that \(\mathrm { P } ( 87 < X < k ) = 0.3\).

6

1. A chess team of 2 girls and 2 boys is to be chosen from the 7 girls and 6 boys in the chess club. Find the number of ways this can be done if 2 of the girls are twins and are either both in the team or both not in the team.
2. 1. The digits of the number 1244687 can be rearranged to give many different 7-digit numbers. How many of these 7 -digit numbers are even?
   2. How many different numbers between 20000 and 30000 can be formed using 5 different digits from the digits \(1,2,4,6,7,8\) ?
3. Helen has some black tiles, some white tiles and some grey tiles. She places a single row of 8 tiles above her washbasin. Each tile she places is equally likely to be black, white or grey. Find the probability that there are no tiles of the same colour next to each other.