Questions — OCR (4907 questions)

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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]
OCR S1 2013 June Q4
6 marks Moderate -0.8
At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m\) kg, and for each person the value of \((m - 5)\) was recorded. The mean and standard deviation of \((m - 5)\) were found to be 0.74 and 0.13 respectively.
  1. Write down the mean and standard deviation of \(m\). [2]
The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.
  1. Calculate the mean of all 25 estimates. [2]
  2. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg, comment on this claim. [2]
OCR S1 2013 June Q5
9 marks Moderate -0.3
The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x\)\%, and the Consumer Price Index (CPI), \(y\)\%, in the United Kingdom from April 2008 to July 2010.
DateApril 2008July 2008October 2008January 2009April 2009July 2009October 2009January 2010April 2010July 2010
UR, \(x\)\%5.25.76.16.87.57.87.87.97.87.7
CPI, \(y\)\%3.04.44.53.02.31.81.53.53.73.1
These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x^2 = 503.45 \quad \sum y = 30.8 \quad \sum y^2 = 103.94 \quad \sum xy = 211.9$$
  1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(-0.6 < r < -0.5\). [3]
  2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement. [1]
    1. Calculate the equation of the regression line of \(x\) on \(y\). [3]
    2. Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%. [2]
OCR S1 2013 June Q6
7 marks Easy -1.3
The diagram shows five cards, each with a letter on it. \includegraphics{figure_6} The letters A and E are vowels; the letters B, C and D are consonants.
  1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them. [2]
  2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it. [3]
  3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it. [2]
OCR S1 2013 June Q7
11 marks Standard +0.3
In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows. • If \(X < 2\), the batch is accepted. • If \(X > 2\), the batch is rejected. • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted. It is given that 5\% of mugs are defective.
    1. Find the probability that the batch is rejected after just the first sample is checked. [3]
    2. Show that the probability that the batch is rejected is 0.327, correct to 3 significant figures. [5]
  2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked. [3]
OCR S1 2013 June Q8
7 marks Moderate -0.3
  1. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours. [3]
  2. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54. Find the number of red discs that were in the second bag at the start. [4]
OCR S1 2013 June Q9
8 marks Standard +0.3
A game is played with a token on a board with a grid printed on it. The token starts at the point \((0, 0)\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8, or 1 unit in the positive \(y\)-direction with probability 0.2. The token stops when it reaches a point with a \(y\)-coordinate of 1. It is given that the token stops at \((X, 1)\).
    1. Find the probability that \(X = 10\). [2]
    2. Find the probability that \(X < 10\). [3]
  2. Find the expected number of steps taken by the token. [2]
  3. Hence, write down the value of E\((X)\). [1]