CAIE S1 (Statistics 1) 2010 June

1 The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$\begin{array} { l l l l l l l } 15 & 10 & 48 & 10 & 19 & 14 & 16 \end{array}$$

1. Find the mean and standard deviation of these times.
2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case.
3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate.

2 The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm .

1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm .
2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm .
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   The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies.

4 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).

1. Given that \(5 \sigma = 3 \mu\), find \(\mathrm { P } ( X < 2 \mu )\).
2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm { P } \left( X < \frac { 1 } { 3 } \mu \right) = 0.8524\). Express \(\mu\) in terms of \(\sigma\).

5 Two fair twelve-sided dice with sides marked \(1,2,3,4,5,6,7,8,9,10,11,12\) are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows.
\(Q\) : the product of the two numbers is 24 .
\(R\) : both of the numbers are greater than 8 .

1. Find \(\mathrm { P } ( Q )\).
2. Find \(\mathrm { P } ( R )\).
3. Are events \(Q\) and \(R\) exclusive? Justify your answer.
4. Are events \(Q\) and \(R\) independent? Justify your answer.

6 A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.

1. Draw up the probability distribution of \(X\).
2. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 7 }\) and calculate \(\operatorname { Var } ( X )\).
3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac { 3 } { 5 }\) or the geese with probability \(\frac { 2 } { 5 }\). If the dog chases the ducks there is a probability of \(\frac { 1 } { 10 }\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac { 3 } { 4 }\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese.

7 Nine cards, each of a different colour, are to be arranged in a line.

1. How many different arrangements of the 9 cards are possible? The 9 cards include a pink card and a green card.
2. How many different arrangements do not have the pink card next to the green card? Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.
3. How many different arrangements in total of 3 cards are possible?
4. How many of the arrangements of 3 cards in part (iii) contain the pink card?
5. How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green card?