OCR S1 (Statistics 1) 2010 June

The marks of some students in a French examination were summarised in a grouped frequency distribution and a cumulative frequency diagram was drawn, as shown below. \includegraphics{figure_1}

1. Estimate how many students took the examination. [1]
2. How can you tell that no student scored more than 55 marks? [1]
3. Find the greatest possible range of the marks. [1]
4. The minimum mark for Grade C was 27. The number of students who gained exactly Grade C was the same as the number of students who gained a grade lower than C. Estimate the maximum mark for Grade C. [3]
5. In a German examination the marks of the same students had an interquartile range of 16 marks. What does this result indicate about the performance of the students in the German examination as compared with the French examination? [3]

Three skaters, \(A\), \(B\) and \(C\), are placed in rank order by four judges. Judge \(P\) ranks skater \(A\) in 1st place, skater \(B\) in 2nd place and skater \(C\) in 3rd place.

1. Without carrying out any calculation, state the value of Spearman's rank correlation coefficient for the following ranks. Give a reason for your answer. [1]

   Skater	\(A\)	\(B\)	\(C\)
   Judge \(P\)	1	2	3
   Judge \(Q\)	3	2	1

2. Calculate the value of Spearman's rank correlation coefficient for the following ranks. [3]

   Skater	\(A\)	\(B\)	\(C\)
   Judge \(P\)	1	2	3
   Judge \(R\)	3	1	2

3. Judge \(S\) ranks the skaters at random. Find the probability that the value of Spearman's rank correlation coefficient between the ranks of judge \(P\) and judge \(S\) is 1. [3]

1. Some values, \((x, y)\), of a bivariate distribution are plotted on a scatter diagram and a regression line is to be drawn. Explain how to decide whether the regression line of \(y\) on \(x\) or the regression line of \(x\) on \(y\) is appropriate. [2]
2. In an experiment the temperature, \(x\) °C, of a rod was gradually increased from 0 °C, and the extension, \(y\), was measured nine times at 50 °C intervals. The results are summarised below. \(n = 9\) \quad \(\Sigma x = 1800\) \quad \(\Sigma y = 14.4\) \quad \(\Sigma x^2 = 510000\) \quad \(\Sigma y^2 = 32.6416\) \quad \(\Sigma xy = 4080\)
   1. Show that the gradient of the regression line of \(y\) on \(x\) is 0.008 and find the equation of this line. [4]
   2. Use your equation to estimate the temperature when the extension is 2.5 mm. [1]
   3. Use your equation to estimate the extension for a temperature of \(-50\) °C. [1]
   4. Comment on the meaning and the reliability of your estimate in part (c). [2]

1. The random variable \(W\) has the distribution B\((10, \frac{1}{4})\). Find
   1. P\((W \leq 2)\), [1]
   2. P\((W = 2)\). [2]
2. The random variable \(X\) has the distribution B\((15, 0.22)\).
   1. Find P\((X = 4)\). [2]
   2. Find E\((X)\) and Var\((X)\). [3]

Each of four cards has a number printed on it as shown. Two of the cards are chosen at random, without replacement. The random variable \(X\) denotes the sum of the numbers on these two cards.

1. Show that P\((X = 6) = \frac{1}{6}\) and P\((X = 4) = \frac{1}{3}\). [3]
2. Write down all the possible values of \(X\) and find the probability distribution of \(X\). [4]
3. Find E\((X)\) and Var\((X)\). [5]

There are 10 numbers in a list. The first 9 numbers have mean 6 and variance 2. The 10th number is 3. Find the mean and variance of all 10 numbers. [6]

The menu below shows all the dishes available at a certain restaurant. A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes and 4 different vegetable dishes from this menu. Given these restrictions,

1. find the number of possible combinations of dishes that they can choose to share, [3]
2. assuming that all choices are equally likely, find the probability that they choose boiled rice. [2]

The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.

1. Find the number of possible combinations of dishes that they can now choose. [3]

The proportion of people who watch West Street on television is 30\%. A market researcher interviews people at random in order to contact viewers of West Street. Each day she has to contact a certain number of viewers of West Street.

1. Near the end of one day she finds that she needs to contact just one more viewer of West Street. Find the probability that the number of further interviews required is
   1. 4, [3]
   2. less than 4. [3]
2. Near the end of another day she finds that she needs to contact just two more viewers of West Street. Find the probability that the number of further interviews required is
   1. 5, [4]
   2. more than 5. [2]