CAIE S1 (Statistics 1) 2014 November

1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$

2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.

1. Find the value of \(k\).
2. Find the mode of \(X\).
3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.

3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:

• if the coin shows a head, the two numbers from the dice are added together;
• if the coin shows a tail, the two numbers from the dice are multiplied together.

Find the probability that the coin shows a head given that Jodie's score is 8 .

4 The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type \(A\) and 43 smartphones of type \(B\).
(7) Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).

1. Find the median and quartiles for smartphones of type \(A\). You are given that the median, lower quartile and upper quartile for smartphones of type \(B\) are 0.46 seconds, 0.36 seconds and 0.63 seconds respectively.
2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
3. Compare the loading times for these two types of smartphone.

5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .

1. Show that the variance of the number of faulty screws in a packet is 1.104 .
2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.

6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .

1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
3. Find the value of \(\mu\).

7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done

1. if there are more women than men on the committee,
2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
3. In how many different ways can the committee members be arranged in a line if the man is not at either end?