Questions — OCR S1 (169 questions)

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OCR S1 Q8
13 marks Moderate -0.3
8 The table shows the population, \(x\) million, of each of nine countries in Western Europe together with the population, \(y\) million, of its capital city.
GermanyUnited KingdomFranceItalySpainThe NetherlandsPortugalAustriaSwitzerland
\(x\)82.159.259.156.739.215.99.98.17.3
\(y\)3.57.09.02.72.90.80.71.60.1
$$\left[ n = 9 , \Sigma x = 337.5 , \Sigma x ^ { 2 } = 18959.11 , \Sigma y = 28.3 , \Sigma y ^ { 2 } = 161.65 , \Sigma x y = 1533.76 . \right]$$
  1. (a) Calculate Spearman's rank correlation coefficient, \(r _ { s }\).
    (b) Explain what your answer indicates about the populations of these countries and their capital cities.
  2. Calculate the product moment correlation coefficient, \(r\). The data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-09_936_881_1162_632}
  3. By considering the diagram, state the effect on the value of the product moment correlation coefficient, \(r\), if the data for France and the United Kingdom were removed from the calculation.
  4. In a certain country in Africa, most people live in remote areas and hence the population of the country is unknown. However, the population of the capital city is known to be approximately 1 million. An official suggests that the population of this country could be estimated by using a regression line drawn on the above scatter diagram.
    (a) State, with a reason, whether the regression line of \(y\) on \(x\) or the regression line of \(x\) on \(y\) would need to be used.
    (b) Comment on the reliability of such an estimate in this situation. 1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$
  5. State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
  6. Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
  7. Find the values of \(\bar { x }\) and \(\bar { y }\). 2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that
  8. the second disc is black, given that the first disc was black,
  9. the second disc is black,
  10. the two discs are of different colours. 3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
  11. How many different arrangements of the letters are possible?
  12. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
  13. Find the probability that at least one of these 2 cards has D printed on it. 4
  14. The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
  15. The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
  16. The random variable \(Z\) has the distribution \(B ( n , 0.27 )\). Find the smallest value of \(n\) such that \(\mathrm { P } ( Z \geqslant 1 ) > 0.95\). 5 The probability distribution of a discrete random variable, \(X\), is given in the table.
    \(x\)0123
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 4 }\)\(p\)\(q\)
    It is given that the expectation, \(\mathrm { E } ( X )\), is \(1 \frac { 1 } { 4 }\).
  17. Calculate the values of \(p\) and \(q\).
  18. Calculate the standard deviation of \(X\).
OCR S1 2010 January Q1
9 marks Moderate -0.8
Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).
  1. State two conditions necessary for \(X\) to have a geometric distribution. [2]
  2. Assuming that \(X\) has the distribution Geo(0.3), find
    1. P\((X = 5)\), [2]
    2. P\((X > 5)\). [3]
  3. Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context. [2]
OCR S1 2010 January Q2
13 marks Moderate -0.8
40 people were asked to guess the length of a certain road. Each person gave their guess, \(l\) km, correct to the nearest kilometre. The results are summarised below.
\(l\)10-1213-1516-2021-30
Frequency113206
    1. Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\). [6]
    2. Explain why your answers are only estimates. [1]
  2. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class. [2]
  3. Explain which class contains the median value of \(l\). [2]
  4. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km. Without calculation state whether the following will increase, decrease or remain the same:
    1. the mean of \(l\), [1]
    2. the standard deviation of \(l\). [1]
OCR S1 2010 January Q3
7 marks Moderate -0.8
The heights, \(h\) m, and weights, \(m\) kg, of five men were measured. The results are plotted on the diagram. \includegraphics{figure_3} The results are summarised as follows. \(n = 5\) \(\Sigma h = 9.02\) \(\Sigma m = 377.7\) \(\Sigma h^2 = 16.382\) \(\Sigma m^2 = 28558.67\) \(\Sigma hm = 681.612\)
  1. Use the summarised data to calculate the value of the product moment correlation coefficient, \(r\). [3]
  2. Comment on your value of \(r\) in relation to the diagram. [2]
  3. It was decided to re-calculate the value of \(r\) after converting the heights to feet and the masses to pounds. State what effect, if any, this will have on the value of \(r\). [1]
  4. One of the men had height 1.63 m and mass 78.4 kg. The data for this man were removed and the value of \(r\) was re-calculated using the original data for the remaining four men. State in general terms what effect, if any, this will have on the value of \(r\). [1]
OCR S1 2010 January Q4
10 marks Moderate -0.3
A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.
\(x\)0123
P\((X = x)\)\(\frac{1}{2}\)\(\frac{1}{4}\)\(\frac{1}{8}\)\(\frac{1}{8}\)
  1. Calculate E\((X)\) and Var\((X)\). [5]
The die is thrown 10 times.
  1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
  2. Find the probability that there are exactly 4 throws on which the score is 2. [3]
OCR S1 2010 January Q5
6 marks Moderate -0.8
A washing-up bowl contains 6 spoons, 5 forks and 3 knives. Three of these 14 items are removed at random, without replacement. Find the probability that
  1. all three items are of different kinds, [3]
  2. all three items are of the same kind. [3]
OCR S1 2010 January Q6
7 marks Standard +0.3
  1. A student calculated the values of the product moment correlation coefficient, \(r\), and Spearman's rank correlation coefficient, \(r_s\), for two sets of bivariate data, \(A\) and \(B\). His results are given below. $$A: \quad r = 0.9 \text{ and } r_s = 1$$ $$B: \quad r = 1 \quad \text{and } r_s = 0.9$$ With the aid of a diagram where appropriate, explain why the student's results for \(A\) could both be correct but his results for \(B\) cannot both be correct. [3]
  2. An old research paper has been partially destroyed. The surviving part of the paper contains the following incomplete information about some bivariate data from an experiment. \includegraphics{figure_6} The mean of \(x\) is 4.5. The equation of the regression line of \(y\) on \(x\) is \(y = 2.4x + 3.7\). The equation of the regression line of \(x\) on \(y\) is \(x = 0.40y\) + [missing constant] Calculate the missing constant at the end of the equation of the second regression line. [4]
OCR S1 2010 January Q7
6 marks Moderate -0.8
The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car.
MaleFemale
Jaguar2515
Bentley128
One member is chosen at random from these 60 members.
  1. Given that this member is male, find the probability that he owns a Jaguar. [2]
Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.
  1. Given that the first one of these members is female, find the probability that both own Jaguars. [4]
OCR S1 2010 January Q8
7 marks Moderate -0.8
The five letters of the word NEVER are arranged in random order in a straight line.
  1. How many different orders of the letters are possible? [2]
  2. In how many of the possible orders are the two Es next to each other? [2]
  3. Find the probability that the first two letters in the order include exactly one letter E. [3]
OCR S1 2010 January Q9
7 marks Standard +0.8
\(R\) and \(S\) are independent random variables each having the distribution Geo\((p)\).
  1. Find P\((R = 1\) and \(S = 1)\) in terms of \(p\). [1]
  2. Show that P\((R = 3\) and \(S = 3) = p^2q^4\), where \(q = 1 - p\). [1]
  3. Use the formula for the sum to infinity of a geometric series to show that $$\text{P}(R = S) = \frac{p}{2-p}.$$ [5]
OCR S1 2013 January Q1
7 marks Moderate -0.8
When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values 2, 4, 6 and 8. The spinner is biased so that P(\(X = x\)) = \(kx\), where \(k\) is a constant.
  1. Show that P(\(X = 6\)) = \(\frac{3}{10}\). [2]
  2. Find E(\(X\)) and Var(\(X\)). [5]
OCR S1 2013 January Q2
6 marks Moderate -0.8
  1. Kathryn is allowed three attempts at a high jump. If she succeeds on any attempt, she does not jump again. The probability that she succeeds on her first attempt is \(\frac{1}{4}\). If she fails on her first attempt, the probability that she succeeds on her second attempt is \(\frac{1}{3}\). If she fails on her first two attempts, the probability that she succeeds on her third attempt is \(\frac{1}{2}\). Find the probability that she succeeds. [3]
  2. Khaled is allowed two attempts to pass an examination. If he succeeds on his first attempt, he does not make a second attempt. The probability that he passes at the first attempt is 0.4 and the probability that he passes on either the first or second attempt is 0.58. Find the probability that he passes on the second attempt, given that he failed on the first attempt. [3]
OCR S1 2013 January Q3
12 marks Moderate -0.3
The Gross Domestic Product per Capita (GDP), \(x\) dollars, and the Infant Mortality Rate per thousand (IMR), \(y\), of 6 African countries were recorded and summarised as follows. \(n = 6\) \quad \(\sum x = 7000\) \quad \(\sum x^2 = 8700000\) \quad \(\sum y = 456\) \quad \(\sum y^2 = 36262\) \quad \(\sum xy = 509900\)
  1. Calculate the equation of the regression line of \(y\) on \(x\) for these 6 countries. [4]
The original data were plotted on a scatter diagram and the regression line of \(y\) on \(x\) was drawn, as shown below. \includegraphics{figure_3}
  1. The GDP for another country, Tanzania, is 1300 dollars. Use the regression line in the diagram to estimate the IMR of Tanzania. [1]
  2. The GDP for Nigeria is 2400 dollars. Give two reasons why the regression line is unlikely to give a reliable estimate for the IMR for Nigeria. [2]
  3. The actual value of the IMR for Tanzania is 96. The data for Tanzania (\(x = 1300, y = 96\)) is now included with the original 6 countries. Calculate the value of the product moment correlation coefficient, \(r\), for all 7 countries. [4]
  4. The IMR is now redefined as the infant mortality rate per hundred instead of per thousand, and the value of \(r\) is recalculated for all 7 countries. Without calculation state what effect, if any, this would have on the value of \(r\) found in part (iv). [1]
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]