OCR S1 (Statistics 1) 2012 June

1 For each of the last five years the number of tourists, \(x\) thousands, visiting Sackton, and the average weekly sales, \(\pounds y\) thousands, in Sackton Stores were noted. The table shows the results.

1. Calculate the product moment correlation coefficient \(r\) between \(x\) and \(y\).
2. It is required to estimate the average weekly sales at Sackton Stores in a year when the number of tourists is 280000 . Calculate the equation of an appropriate regression line, and use it to find this estimate.
3. Over a longer period the value of \(r\) is - 0.8 . The mayor says, "This shows that having more tourists causes sales at Sackton Stores to decrease." Give a reason why this statement is not correct.

2 The masses, \(x \mathrm {~kg}\), of 50 bags of flour were measured and the results were summarised as follows. $$n = 50 \quad \Sigma ( x - 1.5 ) = 1.4 \quad \Sigma ( x - 1.5 ) ^ { 2 } = 0.05$$ Calculate the mean and standard deviation of the masses of these bags of flour.

3 The test marks of 14 students are displayed in a stem-and-leaf diagram, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{e23cb28b-49e5-436a-942d-e6320029c634-2_234_261_1425_482} Key: 1 | 6 means 16 marks

1. Find the lower quartile.
2. Given that the median is 32 , find the values of \(w\) and \(x\).
3. Find the possible values of the upper quartile.
4. State one advantage of a stem-and-leaf diagram over a box-and-whisker plot.
5. State one advantage of a box-and-whisker plot over a stem-and-leaf diagram.

4 A bag contains 5 red discs and 1 black disc. Tina takes two discs from the bag at random without replacement.

1. The diagram shows part of a tree diagram to illustrate this situation. \section*{First disc}
   \includegraphics[max width=\textwidth, alt={}]{e23cb28b-49e5-436a-942d-e6320029c634-3_264_494_479_550}
   Complete the tree diagram in your Answer Book showing all the probabilities. \section*{Second disc}
2. Find the probability that exactly one of the two discs is red. All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.
3. Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.

5

1. Write down the value of Spearman's rank correlation coefficent, \(r _ { s }\), for the following sets of ranks. All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.
2. Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.
   [0pt] [2
3. Write down the value of Spearman's rank correlation coefficent, \(r _ { s }\), for the following sets of ranks.
   (b)

   Judge \(A\) ranks	1	2	3	4
   Judge \(C\) ranks	4	3	2	1

   (a)
   (a)

   Judge \(A\) ranks	1	2	3	4
   Judge \(B\) ranks	1	2	3	4

4. Calculate the value of \(r _ { s }\) for the following ranks.

   Judge \(A\) ranks	1	2	3	4
   Judge \(D\) ranks	2	4	1	3

5. For each of parts (i)(a), (i)(b) and (ii), describe in everyday terms the relationship between the two judges’ opinions.

6 A six-sided die is biased so that the probability of scoring 6 is 0.1 and the probabilities of scoring \(1,2,3,4\), and 5 are all equal. In a game at a fête, contestants pay \(\pounds 3\) to roll this die. If the score is 6 they receive \(\pounds 10\) back. If the score is 5 they receive \(\pounds 5\) back. Otherwise they receive no money back. Find the organiser's expected profit for 100 rolls of the die.

7

1. 5 of the 7 letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) are arranged in a random order in a straight line.
   (a) How many different arrangements of 5 letters are possible?
   (b) How many of these arrangements end with a vowel (A or E)?
2. A group of 5 people is to be chosen from a list of 7 people.
   (a) How many different groups of 5 people can be chosen?
   (b) The list of 7 people includes Jill and Jo. A group of 5 people is chosen at random from the list. Given that either Jill and Jo are both chosen or neither of them is chosen, find the probability that both of them are chosen.

8

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 30,0.6 )\). Find \(\mathrm { P } ( X \geqslant 16 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 4,0.7 )\).
   (a) Find \(\mathrm { P } ( Y = 2 )\).
   (b) Three values of \(Y\) are chosen at random. Find the probability that their total is 10 .

9

1. A clock is designed to chime once each hour, on the hour. The clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 10 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day. Find the probability that the first time it does not chime is
   (a) at 0600 on that day,
   (b) before 0600 on that day.
2. Another clock is designed to chime twice each hour: on the hour and at 30 minutes past the hour. This clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 20 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day.
   (a) Find the probability that the first time it does not chime is at either 0030 or 0130 on that day.
   (b) Use the formula for the sum to infinity of a geometric progression to find the probability that the first time it does not chime is at 30 minutes past some hour.