OCR S1 (Statistics 1) 2009 June

2 Two judges placed 7 dancers in rank order. Both judges placed dancers \(A\) and \(B\) in the first two places, but in opposite orders. The judges agreed about the ranks for all the other 5 dancers. Calculate the value of Spearman's rank correlation coefficient.

5 The diameters of 100 pebbles were measured. The measurements rounded to the nearest millimetre, \(x\), are summarised in the table. These data are to be presented on a statistical diagram.

1. For a histogram, find the frequency density of the \(10 \leqslant x \leqslant 19\) class.
2. For a cumulative frequency graph, state the coordinates of the first two points that should be plotted.
3. Why is it not possible to draw an exact box-and-whisker plot to illustrate the data?

6 Last year Eleanor played 11 rounds of golf. Her scores were as follows: \(79 , \quad 71 , \quad 80 , \quad 67 , \quad 67 , \quad 74 , \quad 66 , \quad 65 , \quad 71 , \quad 66 , \quad 64\).

1. Calculate the mean of these scores and show that the standard deviation is 5.31 , correct to 3 significant figures.
2. Find the median and interquartile range of the scores. This year, Eleanor also played 11 rounds of golf. The standard deviation of her scores was 4.23, correct to 3 significant figures, and the interquartile range was the same as last year.
3. Give a possible reason why the standard deviation of her scores was lower than last year although her interquartile range was unchanged. In golf, smaller scores mean a better standard of play than larger scores. Ken suggests that since the standard deviation was smaller this year, Eleanor's overall standard has improved.
4. Explain why Ken is wrong.
5. State what the smaller standard deviation does show about Eleanor's play.

7 Three letters are selected at random from the 8 letters of the word COMPUTER, without regard to order.

1. Find the number of possible selections of 3 letters.
2. Find the probability that the letter P is included in the selection. Three letters are now selected at random, one at a time, from the 8 letters of the word COMPUTER, and are placed in order in a line.
3. Find the probability that the 3 letters form the word TOP.

8 A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes counters at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white counters have been taken. Niko plays the game once.

1. (a) Copy and complete the tree diagram showing the probabilities for Niko. \section*{First counter} \includegraphics[max width=\textwidth, alt={}, center]{c985b9cc-a202-4d5d-a6b3-591b0560f570-4_293_426_1231_532}
   (b) Find the probability that Niko will win a prize.
2. The number of counters that Niko takes is denoted by \(X\).
   (a) Find \(\mathrm { P } ( X = 3 )\).
   (b) Find \(\mathrm { E } ( X )\).

9 Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is 0.12 .

1. Find the smallest value of \(n\) such that the probability of at least one success in \(n\) trials is more than 0.95.
2. Find the probability that the 3rd success occurs on the 7th trial.