CAIE S1 (Statistics 1) 2010 November

1 Anita made observations of the maximum temperature, \(t ^ { \circ } \mathrm { C }\), on 50 days. Her results are summarised by \(\Sigma t = 910\) and \(\Sigma ( t - \bar { t } ) ^ { 2 } = 876\), where \(\bar { t }\) denotes the mean of the 50 observations. Calculate \(\bar { t }\) and the standard deviation of the observations.

2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.

3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.

1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.

4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table. A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .

1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
2. Calculate an estimate of the mean weight of the stones.

5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, \(A , B , C\) and \(D\). Each friend chooses an entrance independently.

• The probability that Rick chooses entrance \(A\) is \(\frac { 1 } { 3 }\). The probabilities that he chooses entrances \(B , C\) or \(D\) are all equal.
• Brenda is equally likely to choose any of the four entrances.
• The probability that Ali chooses entrance \(C\) is \(\frac { 2 } { 7 }\) and the probability that he chooses entrance \(D\) is \(\frac { 3 } { 5 }\). The probabilities that he chooses the other two entrances are equal.
  1. Find the probability that at least 2 friends will choose entrance \(B\).
  2. Find the probability that the three friends will all choose the same entrance.

6
\includegraphics[max width=\textwidth, alt={}, center]{fcf7b1c6-cc76-4c84-998c-9de6a7e9bb2d-3_163_618_260_765} Pegs are to be placed in the four holes shown, one in each hole. The pegs come in different colours and pegs of the same colour are identical. Calculate how many different arrangements of coloured pegs in the four holes can be made using

1. 6 pegs, all of different colours,
2. 4 pegs consisting of 2 blue pegs, 1 orange peg and 1 yellow peg. Beryl has 12 pegs consisting of 2 red, 2 blue, 2 green, 2 orange, 2 yellow and 2 black pegs. Calculate how many different arrangements of coloured pegs in the 4 holes Beryl can make using
3. 4 different colours,
4. 3 different colours,
5. any of her 12 pegs.

7 Sanket plays a game using a biased die which is twice as likely to land on an even number as on an odd number. The probabilities for the three even numbers are all equal and the probabilities for the three odd numbers are all equal.

1. Find the probability of throwing an odd number with this die. Sanket throws the die once and calculates his score by the following method.
   • If the number thrown is 3 or less he multiplies the number thrown by 3 and adds 1 .
   • If the number thrown is more than 3 he multiplies the number thrown by 2 and subtracts 4 .
   The random variable \(X\) is Sanket's score.
2. Show that \(\mathrm { P } ( X = 8 ) = \frac { 2 } { 9 }\). The table shows the probability distribution of \(X\).

   \(x\)	4	6	7	8	10
   \(\mathrm { P } ( X = x )\)	\(\frac { 3 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)

3. Given that \(\mathrm { E } ( X ) = \frac { 58 } { 9 }\), find \(\operatorname { Var } ( X )\). Sanket throws the die twice.
4. Find the probability that the total of the scores on the two throws is 16 .
5. Given that the total of the scores on the two throws is 16 , find the probability that the score on the first throw was 6 .