CAIE S1 (Statistics 1) 2017 March

1 Twelve values of \(x\) are shown below. Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]

2 A bag contains 10 pink balloons, 9 yellow balloons, 12 green balloons and 9 white balloons. 7 balloons are selected at random without replacement. Find the probability that exactly 3 of them are green.

3 It is found that \(10 \%\) of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.

4 The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data. Key: 748 | 5 represents 0.7485 kg .

1. On the grid, draw a box-and-whisker plot to represent the data.
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2. Name a distribution that might be a suitable model for the weights of this type of cereal packet. Justify your answer.

5

1. A plate of cakes holds 12 different cakes. Find the number of ways these cakes can be shared between Alex and James if each receives an odd number of cakes.
2. Another plate holds 7 cup cakes, each with a different colour icing, and 4 brownies, each of a different size. Find the number of different ways these 11 cakes can be arranged in a row if no brownie is next to another brownie.
3. A plate of biscuits holds 4 identical chocolate biscuits, 6 identical shortbread biscuits and 2 identical gingerbread biscuits. These biscuits are all placed in a row. Find how many different arrangements are possible if the chocolate biscuits are all kept together.

6 Pack \(A\) consists of ten cards numbered \(0,0,1,1,1,1,1,3,3,3\). Pack \(B\) consists of six cards numbered \(0,0,2,2,2,2\). One card is chosen at random from each pack. The random variable \(X\) is defined as the sum of the two numbers on the cards.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 2 } { 15 }\).
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2. Draw up the probability distribution table for \(X\).
3. Given that \(X = 3\), find the probability that the card chosen from pack \(A\) is a 1 .

7

1. The lengths, in centimetres, of middle fingers of women in Raneland have a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(25 \%\) of these women have fingers longer than 8.8 cm and \(17.5 \%\) have fingers shorter than 7.7 cm .
   1. Find the values of \(\mu\) and \(\sigma\).
      The lengths, in centimetres, of middle fingers of women in Snoland have a normal distribution with mean 7.9 and standard deviation 0.44. A random sample of 5 women from Snoland is chosen.
   2. Find the probability that exactly 3 of these women have middle fingers shorter than 8.2 cm .
2. The random variable \(X\) has a normal distribution with mean equal to the standard deviation. Find the probability that a particular value of \(X\) is less than 1.5 times the mean.