Questions S1 (2020 questions)

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OCR S1 2013 January Q4
10 marks Moderate -0.8
  1. How many different 3-digit numbers can be formed using the digits 1, 2 and 3 when
    1. no repetitions are allowed, [1]
    2. any repetitions are allowed, [2]
    3. each digit may be included at most twice? [2]
  2. How many different 4-digit numbers can be formed using the digits 1, 2 and 3 when each digit may be included at most twice? [5]
OCR S1 2013 January Q5
10 marks Moderate -0.8
A random variable \(X\) has the distribution B\((5, \frac{1}{4})\).
  1. Find
    1. E(\(X\)), [1]
    2. P(\(X = 2\)). [2]
  2. Two values of \(X\) are chosen at random. Find the probability that their sum is less than 2. [4]
  3. 10 values of \(X\) are chosen at random. Use an appropriate formula to find the probability that exactly 3 of these values are 2s. [3]
OCR S1 2013 January Q6
7 marks Moderate -0.8
The masses, \(x\) grams, of 800 apples are summarised in the histogram. \includegraphics{figure_6}
  1. On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\). [3]
  2. Find an estimate of the median mass of the apples. [4]
OCR S1 2013 January Q7
7 marks Standard +0.3
  1. Two judges rank \(n\) competitors, where \(n\) is an even number. Judge 2 reverses each consecutive pair of ranks given by Judge 1, as shown.
    Competitor\(C_1\)\(C_2\)\(C_3\)\(C_4\)\(C_5\)\(C_6\)\(\ldots\)\(C_{n-1}\)\(C_n\)
    Judge 1 rank123456\(\ldots\)\(n-1\)\(n\)
    Judge 2 rank214365\(\ldots\)\(n\)\(n-1\)
    Given that the value of Spearman's coefficient of rank correlation is \(\frac{63}{65}\), find \(n\). [4]
  2. An experiment produced some data from a bivariate distribution. The product moment correlation coefficient is denoted by \(r\), and Spearman's rank correlation coefficient is denoted by \(r_s\).
    1. Explain whether the statement $$r = 1 \Rightarrow r_s = 1$$ is true or false. [1]
    2. Use a diagram to explain whether the statement $$r \neq 1 \Rightarrow r_s \neq 1$$ is true or false. [2]
OCR S1 2013 January Q8
13 marks Standard +0.3
Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1.
  1. Find the probability that
    1. the first time she succeeds is on her 5th attempt, [2]
    2. the first time she succeeds is after her 5th attempt, [2]
    3. the second time she succeeds is before her 4th attempt. [4]
    Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2. Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  2. Find the probability that the first person to hit the target is Sandra, on her
    1. 2nd attempt, [2]
    2. 10th attempt. [3]
OCR S1 2009 June Q1
7 marks Easy -1.2
20% of packets of a certain kind of cereal contain a free gift. Jane buys one packet a week for 8 weeks. The number of free gifts that Jane receives is denoted by \(X\). Assuming that Jane's 8 packets can be regarded as a random sample, find
  1. P(\(X = 3\)), [3]
  2. P(\(X \geqslant 3\)), [2]
  3. E(\(X\)). [2]
OCR S1 2009 June Q2
4 marks Moderate -0.8
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. [4]
OCR S1 2009 June Q3
8 marks Moderate -0.3
In an agricultural experiment, the relationship between the amount of water supplied, \(x\) units, and the yield, \(y\) units, was investigated. Six values of \(x\) were chosen and for each value of \(x\) the corresponding value of \(y\) was measured. The results are shown in the table.
\(x\)123456
\(y\)36881110
These results, together with the regression line of \(y\) on \(x\), are plotted on the graph. \includegraphics{figure_1}
  1. Give a reason why the regression line of \(x\) on \(y\) is not suitable in this context. [1]
  2. Explain the significance, for the regression line of \(y\) on \(x\), of the distances shown by the vertical dotted lines in the diagram. [2]
  3. Calculate the value of the product moment correlation coefficient, \(r\). [3]
  4. Comment on your value of \(r\) in relation to the diagram. [2]
OCR S1 2009 June Q4
8 marks Moderate -0.8
30% of people own a Talk-2 phone. People are selected at random, one at a time, and asked whether they own a Talk-2 phone. The number of people questioned, up to and including the first person who owns a Talk-2 phone, is denoted by \(X\). Find
  1. P(\(X = 4\)), [3]
  2. P(\(X > 4\)), [2]
  3. P(\(X < 6\)). [3]
OCR S1 2009 June Q5
5 marks Moderate -0.8
The diameters of 100 pebbles were measured. The measurements rounded to the nearest millimetre, \(x\), are summarised in the table.
\(x\)\(10 \leqslant x \leqslant 19\)\(20 \leqslant x \leqslant 24\)\(25 \leqslant x \leqslant 29\)\(30 \leqslant x \leqslant 49\)
Number of stones25222924
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]
  2. For a cumulative frequency graph, state the coordinates of the first two points that should be plotted. [2]
  3. Why is it not possible to draw an exact box-and-whisker plot to illustrate the data? [1]
OCR S1 2009 June Q6
11 marks Moderate -0.8
Last year Eleanor played 11 rounds of golf. Her scores were as follows: 79, 71, 80, 67, 67, 74, 66, 65, 71, 66, 64.
  1. Calculate the mean of these scores and show that the standard deviation is 5.31, correct to 3 significant figures. [4]
  2. Find the median and interquartile range of the scores. [4]
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.
  1. Give a possible reason why the standard deviation of her scores was lower than last year although her interquartile range was unchanged. [1]
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.
  1. Explain why Ken is wrong. [1]
  2. State what the smaller standard deviation does show about Eleanor's play. [1]
OCR S1 2009 June Q7
8 marks Moderate -0.8
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]
  2. Find the probability that the letter P is included in the selection. [3]
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.
  1. Find the probability that the 3 letters form the word TOP. [3]
OCR S1 2009 June Q8
13 marks Moderate -0.3
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. Copy and complete the tree diagram showing the probabilities for Niko. [4] \includegraphics{figure_2}
    2. Find the probability that Niko will win a prize. [3]
  2. The number of counters that Niko takes is denoted by \(X\).
    1. Find P(\(X = 3\)). [2]
    2. Find E(\(X\)). [4]
OCR S1 2009 June Q9
8 marks Standard +0.3
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. [3]
  2. Find the probability that the 3rd success occurs on the 7th trial. [5]
OCR S1 2010 June Q1
9 marks Easy -1.2
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]
OCR S1 2010 June Q2
7 marks Moderate -0.8
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 \(P\)123
    Judge \(Q\)321
  2. Calculate the value of Spearman's rank correlation coefficient for the following ranks. [3]
    Skater\(A\)\(B\)\(C\)
    Judge \(P\)123
    Judge \(R\)312
  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]
OCR S1 2010 June Q3
10 marks Moderate -0.8
  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]
OCR S1 2010 June Q4
8 marks Easy -1.3
  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]
OCR S1 2010 June Q5
12 marks Moderate -0.8
Each of four cards has a number printed on it as shown.
1233
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]
OCR S1 2010 June Q6
6 marks Moderate -0.8
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]
OCR S1 2010 June Q7
8 marks Moderate -0.8
The menu below shows all the dishes available at a certain restaurant.
Rice dishesMain dishesVegetable dishes
Boiled riceChickenMushrooms
Fried riceBeefCauliflower
Pilau riceLambSpinach
Keema riceMixed grillLentils
PrawnPotatoes
Vegetarian
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]
OCR S1 2010 June Q8
12 marks Moderate -0.3
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]
OCR S1 2013 June Q1
7 marks Easy -1.8
The lengths, in centimetres, of 18 snakes are given below. 24 62 20 65 27 67 69 32 40 53 55 47 33 45 55 56 49 58
  1. Draw an ordered stem-and-leaf diagram for the data. [3]
  2. Find the mean and median of the lengths of the snakes. [2]
  3. It was found that one of the lengths had been measured incorrectly. After this length was corrected, the median increased by 1 cm. Give two possibilities for the incorrect length and give a corrected value in each case. [2]
OCR S1 2013 June Q2
7 marks Moderate -0.8
  1. The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.
    StudentAnnBillCazDenEd
    Time revising0603510045
    GradeCDEBA
    Calculate Spearman's rank correlation coefficient. [5]
  2. The table below shows the ranks given by two judges to four competitors.
    CompetitorPQRS
    Judge 1 rank1234
    Judge 2 rank3214
    Spearman's rank correlation coefficient for these ranks is denoted by \(r_s\). With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of \(r_s\). There is no need to find the value of \(r_s\). [2]
OCR S1 2013 June Q3
10 marks Moderate -0.8
The probability distribution of a random variable \(X\) is shown.
\(x\)1357
P\((X = x)\)0.40.30.20.1
  1. Find E\((X)\) and Var\((X)\). [5]
  2. Three independent values of \(X\), denoted by \(X_1\), \(X_2\) and \(X_3\), are chosen. Given that \(X_1 + X_2 + X_3 = 19\), write down all the possible sets of values for \(X_1\), \(X_2\) and \(X_3\) and hence find P\((X_1 = 7)\). [2]
  3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5s. [3]