CAIE S1 (Statistics 1) 2010 November

1 The discrete random variable \(X\) takes the values 1, 4, 5, 7 and 9 only. The probability distribution of \(X\) is shown in the table. Find \(p\).

2 Esme noted the test marks, \(x\), of 16 people in a class. She found that \(\Sigma x = 824\) and that the standard deviation of \(x\) was 6.5.

1. Calculate \(\Sigma ( x - 50 )\) and \(\Sigma ( x - 50 ) ^ { 2 }\).
2. One person did the test later and her mark was 72. Calculate the new mean and standard deviation of the marks of all 17 people.

3 A fair five-sided spinner has sides numbered 1,2,3,4,5. Raj spins the spinner and throws two fair dice. He calculates his score as follows.

• If the spinner lands on an even-numbered side, Raj multiplies the two numbers showing on the dice to get his score.
• If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to get his score.

Given that Raj's score is 12, find the probability that the spinner landed on an even-numbered side.

4 The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows.
Sugar: 1.9611 .98312 .00812 .0141 .9681 .9941 .2 .0112 .0171 .9771 .9841 .989
Flour: \(\begin{array} { l l l l l l l } 1.945 & 1.962 & 1.949 & 1.977 & 1.964 & 1.941 & 1.953 \end{array}\)

1. Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand side.
2. Find the median and interquartile range of the weights of the bags of sugar.

5 The distance the Zotoc car can travel on 20 litres of fuel is normally distributed with mean 320 km and standard deviation 21.6 km . The distance the Ganmor car can travel on 20 litres of fuel is normally distributed with mean 350 km and standard deviation 7.5 km . Both cars are filled with 20 litres of fuel and are driven towards a place 367 km away.

1. For each car, find the probability that it runs out of fuel before it has travelled 367 km .
2. The probability that a Zotoc car can travel at least \(( 320 + d ) \mathrm { km }\) on 20 litres of fuel is 0.409 . Find the value of \(d\).

6

1. State three conditions that must be satisfied for a situation to be modelled by a binomial distribution. On any day, there is a probability of 0.3 that Julie's train is late.
2. Nine days are chosen at random. Find the probability that Julie's train is late on more than 7 days or fewer than 2 days.
3. 90 days are chosen at random. Find the probability that Julie's train is late on more than 35 days or fewer than 27 days.

7 A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women.

1. Find the number of possible committees that can be chosen.
2. Find the probability that one particular man, Albert, and one particular woman, Tracey, are both on the committee.
3. Find the number of possible committees that include either Albert or Tracey but not both.
4. The committee that is chosen consists of 4 men and 2 women. They queue up randomly in a line for refreshments. Find the probability that the women are not next to each other in the queue.