CAIE S1 (Statistics 1) 2018 June

1 In a statistics lesson 12 people were asked to think of a number, \(x\), between 1 and 20 inclusive. From the results Tom found that \(\Sigma x = 186\) and that the standard deviation of \(x\) is 4.5. Assuming that Tom's calculations are correct, find the values of \(\Sigma ( x - 10 )\) and \(\Sigma ( x - 10 ) ^ { 2 }\).

2 In a survey 55 students were asked to record, to the nearest kilometre, the total number of kilometres they travelled to school in a particular week. The results are shown below.

1. On the grid, draw a box-and-whisker plot to illustrate the data.
   \includegraphics[max width=\textwidth, alt={}, center]{246c92f4-7603-43ff-8533-042a4be99a69-04_512_1596_900_262} 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.

3 Andy has 4 red socks and 8 black socks in his drawer. He takes 2 socks at random from his drawer.

1. Find the probability that the socks taken are of different colours.
   The random variable \(X\) is the number of red socks taken.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

4

1. The distance that car tyres of a certain make can travel before they need to be replaced has a normal distribution. A survey of a large number of these tyres found that the probability of this distance being more than 36800 km is 0.0082 and the probability of this distance being more than 31000 km is 0.6915 . Find the mean and standard deviation of the distribution.
2. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(3 \sigma = 4 \mu\) and \(\mu \neq 0\). Find \(\mathrm { P } ( X < 3 \mu )\). [3]

5 In Pelmerdon 22\% of families own a dishwasher.

1. Find the probability that, of 15 families chosen at random from Pelmerdon, between 4 and 6 inclusive own a dishwasher.
2. A random sample of 145 families from Pelmerdon is chosen. Use a suitable approximation to find the probability that more than 26 families own a dishwasher.

6 Vehicles approaching a certain road junction from town \(A\) can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town \(A\), \(55 \%\) turn left, \(15 \%\) turn right and \(30 \%\) go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.

1. Find the probability that, of the next three vehicles approaching the junction from town \(A\), one goes straight on and the other two either both turn left or both turn right.
2. Three vehicles approach the junction from town \(A\). Given that all three drivers choose the same direction at the junction, find the probability that they all go straight on.

7 Find the number of different ways in which all 9 letters of the word MINCEMEAT can be arranged in each of the following cases.

1. There are no restrictions.
2. No vowel (A, E, I are vowels) is next to another vowel.
   5 of the 9 letters of the word MINCEMEAT are selected.
3. Find the number of possible selections which contain exactly 1 M and exactly 1 E .
4. Find the number of possible selections which contain at least 1 M and at least 1 E .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.