CAIE S1 (Statistics 1) 2015 June

1 A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3 and 5 inclusive. 2120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table. Calculate estimates of the mean and standard deviation of the reading times.

3
\includegraphics[max width=\textwidth, alt={}, center]{b4bd1629-e2ae-4395-9773-4b14ce428ca6-2_1147_1182_884_477} In an open-plan office there are 88 computers. The times taken by these 88 computers to access a particular web page are represented in the cumulative frequency diagram.

1. On graph paper draw a box-and-whisker plot to summarise this information. An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
2. Show that there are no outliers.
   \includegraphics[max width=\textwidth, alt={}, center]{b4bd1629-e2ae-4395-9773-4b14ce428ca6-3_451_1561_258_292} Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3 , or a handbag. The probability that her mother will like the choice of scarf is 0.72 . The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783 .
3. Find \(x\).

(ii) Given that Nikita's mother does not like her present, find the probability that the present is a scarf.

5 A box contains 5 discs, numbered 1, 2, 4, 6, 7. William takes 3 discs at random, without replacement, and notes the numbers on the discs.

1. Find the probability that the numbers on the 3 discs are two even numbers and one odd number. The smallest of the numbers on the 3 discs taken is denoted by the random variable \(S\).
2. By listing all possible selections (126, 246 and so on) draw up the probability distribution table for \(S\).

6

1. Find the number of different ways the 7 letters of the word BANANAS can be arranged
   1. if the first letter is N and the last letter is B ,
   2. if all the letters A are next to each other.
2. Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together.

7

1. Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
   1. Find the expected number of days during a year ( 52 weeks) for which Zak takes less than 30 minutes for his run.
   2. The probability that Zak's time is between 35.2 minutes and \(t\) minutes, where \(t > 35.2\), is 0.148 . Find the value of \(t\).
2. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 7 ) = 0.2119\) and \(\mathrm { P } ( X < 10 ) = 0.6700\). Find the values of \(\mu\) and \(\sigma\).