CAIE S1 (Statistics 1) 2013 November

1 The distance of a student's home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table. Dominic is asked to draw a histogram to illustrate the data. Dominic's diagram is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{d6836b62-75e7-410e-ab1e-83c391b85948-2_1225_1303_628_422} Give two reasons why this is not a correct histogram.

2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .

3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.

1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).

4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .

1. Find the mean weight of the 45 fruits.
2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.

5

1. The random variable \(X\) is normally distributed with mean 82 and standard deviation 7.4. Find the value of \(q\) such that \(\mathrm { P } ( 82 - q < X < 82 + q ) = 0.44\).
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(5 \mu = 2 \sigma ^ { 2 }\) and that \(\mathrm { P } \left( Y < \frac { 1 } { 2 } \mu \right) = 0.281\). Find the values of \(\mu\) and \(\sigma\).

6

1. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is \(R\).
2. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.
3. The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.
4. Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.

7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.

1. Find the probability that the two digits chosen are equal.
2. Find the probability that one digit is a 5 and one digit is not a 5 .
3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).