CAIE S1 (Statistics 1) 2013 June

1 A summary of 30 values of \(x\) gave the following information: $$\Sigma ( x - c ) = 234 , \quad \Sigma ( x - c ) ^ { 2 } = 1957.5 ,$$ where \(c\) is a constant.

1. Find the standard deviation of these values of \(x\).
2. Given that the mean of these values is 86 , find the value of \(c\).

2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.

3 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males. Key: 2 | 20 | 3 means \\(20200 for females and
)20300 for males.

1. Find the median and the quartiles of the females' salaries. You are given that the median salary of the males is \(
   ) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(
   ) 25300$.
2. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.

4

1. The random variable \(Y\) is normally distributed with positive mean \(\mu\) and standard deviation \(\frac { 1 } { 2 } \mu\). Find the probability that a randomly chosen value of \(Y\) is negative.
2. The weights of bags of rice are normally distributed with mean 2.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 8000 such bags, 253 weighed over 2.1 kg . Find the value of \(\sigma\). [4]

5 Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The random variable \(X\) is the number of these 12 integers which are multiples of 5 .

1. State the distribution of \(X\) and give its parameters.
2. Calculate the probability that \(X\) is between 3 and 5 inclusive. Fiona now produces \(n\) random integers between 7 and 21 inclusive.
3. Find the least possible value of \(n\) if the probability that none of these integers is a multiple of 5 is less than 0.01.

6 Four families go to a theme park together. Mr and Mrs Lin take their 2 children. Mr O'Connor takes his 2 children. Mr and Mrs Ahmed take their 3 children. Mrs Burton takes her son. The 14 people all have to go through a turnstile one at a time to enter the theme park.

1. In how many different orders can the 14 people go through the turnstile if each family stays together?
2. In how many different orders can the 8 children and 6 adults go through the turnstile if no two adults go consecutively? Once inside the theme park, the children go on the roller-coaster. Each roller-coaster car holds 3 people.
3. In how many different ways can the 8 children be divided into two groups of 3 and one group of 2 to go on the roller-coaster?

7 Box \(A\) contains 8 white balls and 2 yellow balls. Box \(B\) contains 5 white balls and \(x\) yellow balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen. \begin{figure}[h] \end{figure}

1. Justify the probability \(\frac { x } { x + 6 }\) on the tree diagram.
2. Copy and complete the tree diagram.
3. If the ball chosen from box \(A\) is white then the probability that the ball chosen from box \(B\) is also white is \(\frac { 1 } { 3 }\). Show that the value of \(x\) is 12 .
4. Given that the ball chosen from box \(B\) is yellow, find the conditional probability that the ball chosen from box \(A\) was yellow.