CAIE S1 (Statistics 1) 2016 June

1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .

1. Copy and complete the following table to show the number of adults in each category.

   	Wears spectacles	Does not wear spectacles	Total
   Right-handed
   Not right-handed
   Total			30

   An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.

2 A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.

1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
2. State two comparisons of the times taken by girls and boys.

3 Two ordinary fair dice are thrown. The resulting score is found as follows.

• If the two dice show different numbers, the score is the smaller of the two numbers.
• If the two dice show equal numbers, the score is 0 .
  1. Draw up the probability distribution table for the score.
  2. Calculate the expected score.

4 The monthly rental prices, \(
) x$, for 9 apartments in a certain city are listed and are summarised as follows. $$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$ The mean monthly rental price is \(
) 2205$.

1. Find the value of the constant \(c\).
2. Find the variance of these values of \(x\).
3. Another apartment is added to the list. The mean monthly rental price is now \(
   ) 2120.50$. Find the rental price of this additional apartment.

5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .

1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
2. Calculate an estimate of the number of these desks that are comfortable for Jodu.

6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if

1. there are no restrictions,
2. the first letter is R and the last letter is G ,
3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
4. Find the number of selections which contain no Es and exactly 1 R .
5. Find the number of selections which contain no Es.

7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.

1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.