Questions — OCR S1 (169 questions)

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OCR S1 2016 June Q1
8 marks Moderate -0.3
1 The table shows the probability distribution of a random variable \(X\).
\(x\)1234
\(\mathrm { P } ( X = x )\)0.10.30.40.2
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. Three values of \(X\) are chosen at random. Find the probability that \(X\) takes the value 2 at least twice.
OCR S1 2016 June Q2
10 marks Moderate -0.3
2
  1. The table shows the amount, \(x\), in hundreds of pounds, spent on heating and the number of absences, \(y\), at a factory during each month in 2014.
    Amount, \(x\), spent on
    heating (£ hundreds)
    		212319151452109201823
    Number of absences, \(y\)2325181812104911152026
    \(n = 12 \quad \Sigma x = 179 \quad \Sigma x ^ { 2 } = 3215 \quad \Sigma y = 191 \quad \Sigma y ^ { 2 } = 3565 \quad \Sigma x y = 3343\)
    1. Calculate \(r\), the product moment correlation coefficient, showing that \(r > 0.92\).
    2. A manager says, 'The value of \(r\) shows that spending more money on heating causes more absences, so we should spend less on heating.' Comment on this claim.
    3. The months in 2014 were numbered \(1,2,3 , \ldots , 12\). The output, \(z\), in suitable units was recorded along with the month number, \(n\), for each month in 2014. The equation of the regression line of \(z\) on \(n\) was found to be \(z = 0.6 n + 17\).
      (a) Use this equation to explain whether output generally increased or decreased over these months.
      (b) Find the mean of \(n\) and use the equation of the regression line to calculate the mean of \(z\).
    4. Hence calculate the total output in 2014.
OCR S1 2016 June Q3
13 marks Moderate -0.8
3 The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$
  1. Find the mean and variance of the masses of these 52 apples.
  2. Use your answers from part (i) to find the exact value of \(\Sigma m ^ { 2 }\). The masses of the apples are illustrated in the box-and-whisker plot below. \includegraphics[max width=\textwidth, alt={}, center]{b5ce3230-7528-439c-9e85-ef159a49cba3-3_250_1310_662_383}
  3. How many apples have masses in the interval \(130 \leqslant m < 140\) ?
  4. An 'outlier' is a data item that lies more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. Explain whether any of the masses of these apples are outliers.
OCR S1 2016 June Q4
8 marks Moderate -0.3
4 In this question the product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
  1. The scatter diagram in Fig. 1 shows the results of an experiment involving some bivariate data. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_597_595_434_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Write down the value of \(r _ { s }\) for these data.
  2. On the diagram in the Answer Booklet, draw five points such that \(r _ { s } = 1\) and \(r \neq 1\).
  3. The scatter diagram in Fig. 2 shows the results of another experiment involving 5 items of bivariate data. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_604_608_1484_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Calculate the value of \(r _ { s }\).
  4. A random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.6 )\). Find
    1. \(\mathrm { P } ( X \leqslant 14 )\),
    2. \(\mathrm { P } ( X = 14 )\),
    3. \(\quad \operatorname { Var } ( X )\).
    4. A random variable \(Y\) has the distribution \(\mathrm { B } ( 24,0.3 )\). Write down an expression for \(\mathrm { P } ( Y = y )\) and evaluate this probability in the case where \(y = 8\).
    5. A random variable \(Z\) has the distribution \(\mathrm { B } ( 2,0.2 )\). Find the probability that two randomly chosen values of \(Z\) are equal.
      (a) Find the number of ways in which 12 people can be divided into three groups containing 5 people, 4 people and 3 people, without regard to order.
      (b) The diagram shows 7 cards, each with a letter on it. $$\mathrm { A } \mathrm {~A} \mathrm {~A} \mathrm {~B} \text { } \mathrm { B } \text { } \mathrm { R } \text { } \mathrm { R }$$ The 7 cards are arranged in a random order in a straight line.
      1. Find the number of possible arrangements of the 7 letters.
      2. Find the probability that the 7 letters form the name BARBARA. The 7 cards are shuffled. Now 4 of the 7 cards are chosen at random and arranged in a random order in a straight line.
      3. Find the probability that the letters form the word ABBA .
OCR S1 Specimen Q1
5 marks Easy -1.2
1 Janet and John wanted to compare their daily journey times to work, so they each kept a record of their journey times for a few weeks.
  1. Janet's daily journey times, \(x\) minutes, for a period of 25 days, were summarised by \(\Sigma x = 2120\) and \(\Sigma x ^ { 2 } = 180044\). Calculate the mean and standard deviation of Janet's journey times.
  2. John's journey times had a mean of 79.7 minutes and a standard deviation of 6.22 minutes. Describe briefly, in everyday terms, how Janet and John's journey times compare.
OCR S1 Specimen Q2
7 marks Standard +0.3
2 Two independent assessors awarded marks to each of 5 projects. The results were as shown in the table.
Project\(A\)\(B\)\(C\)\(D\)\(E\)
First assessor3891628361
Second assessor5684418562
  1. Calculate Spearman's rank correlation coefficient for the data.
  2. Show, by sketching a suitable scatter diagram, how two assessors might have assessed 5 projects in such a way that Spearman's rank correlation coefficient for their marks was + 1 while the product moment correlation coefficient for their marks was not + 1 . (Your scatter diagram need not be drawn accurately to scale.)
OCR S1 Specimen Q3
8 marks Standard +0.3
3 Five friends, Ali, Bev, Carla, Don and Ed, stand in a line for a photograph.
  1. How many different possible arrangements are there if Ali, Bev and Carla stand next to each other?
  2. How many different possible arrangements are there if none of Ali, Bev and Carla stand next to each other?
  3. If all possible arrangements are equally likely, find the probability that two of Ali, Bev and Carla are next to each other, but the third is not next to either of the other two.
OCR S1 Specimen Q4
8 marks Moderate -0.8
4 Each packet of the breakfast cereal Fizz contains one plastic toy animal. There are five different animals in the set, and the cereal manufacturers use equal numbers of each. Without opening a packet it is impossible to tell which animal it contains. A family has already collected four different animals at the start of a year and they now need to collect an elephant to complete their set. The family is interested in how many packets they will need to buy before they complete their set.
  1. Name an appropriate distribution with which to model this situation. State the value(s) of any parameter(s) of the distribution, and state also any assumption(s) needed for the distribution to be a valid model.
  2. Find the probability that the family will complete their set with the third packet they buy after the start of the year.
  3. Find the probability that, in order to complete their collection, the family will need to buy more than 4 packets after the start of the year.
OCR S1 Specimen Q5
10 marks Moderate -0.8
5 A sixth-form class consists of 7 girls and 5 boys. Three students from the class are chosen at random. The number of boys chosen is denoted by the random variable \(X\). Show that
  1. \(\quad \mathrm { P } ( X = 0 ) = \frac { 7 } { 44 }\),
  2. \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 22 }\). The complete probability distribution of \(X\) is shown in the following table.
    \(x\)0123
    \(\mathrm { P } ( X = x )\)\(\frac { 7 } { 44 }\)\(\frac { 21 } { 44 }\)\(\frac { 7 } { 22 }\)\(\frac { 1 } { 22 }\)
  3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 Specimen Q6
11 marks Moderate -0.5
6 \includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-3_803_1180_1018_413} The diagram shows the cumulative frequency graphs for the marks scored by the candidates in an examination. The 2000 candidates each took two papers; the upper curve shows the distribution of marks on paper 1 and the lower curve shows the distribution on paper 2. The maximum mark on each paper was 100.
  1. Use the diagram to estimate the median mark for each of paper 1 and paper 2.
  2. State with a reason which of the two papers you think was the easier one.
  3. To achieve grade A on paper 1 candidates had to score 66 marks out of 100. What mark on paper 2 gives equal proportions of candidates achieving grade A on the two papers? What is this proportion?
  4. The candidates' marks for the two papers could also be illustrated by means of a pair of box-and whisker plots. Give two brief comments comparing the usefulness of cumulative frequency graphs and box-and-whisker plots for representing the data.
OCR S1 Specimen Q7
10 marks Moderate -0.3
7 Items from a production line are examined for any defects. The probability that any item will be found to be defective is 0.15 , independently of all other items.
  1. A batch of 16 items is inspected. Using tables of cumulative binomial probabilities, or otherwise, find the probability that
    1. at least 4 items in the batch are defective,
    2. exactly 4 items in the batch are defective.
    3. Five batches, each containing 16 items, are taken.
      (a) Find the probability that at most 2 of these 5 batches contain at least 4 defective items.
      (b) Find the expected number of batches that contain at least 4 defective items.
OCR S1 Specimen Q8
13 marks Moderate -0.8
8 An experiment was conducted to see whether there was any relationship between the maximum tidal current, \(y \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), and the tidal range, \(x\) metres, at a particular marine location. [The tidal range is the difference between the height of high tide and the height of low tide.] Readings were taken over a period of 12 days, and the results are shown in the following table.
\(x\)2.02.43.03.13.43.73.83.94.04.54.64.9
\(y\)15.222.025.233.033.134.251.042.345.050.761.059.2
$$\left[ \Sigma x = 43.3 , \Sigma y = 471.9 , \Sigma x ^ { 2 } = 164.69 , \Sigma y ^ { 2 } = 20915.75 , \Sigma x y = 1837.78 . \right]$$ The scatter diagram below illustrates the data. \includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-4_462_793_1464_644}
  1. Calculate the product moment correlation coefficient for the data, and comment briefly on your answer with reference to the appearance of the scatter diagram.
  2. Calculate the equation of the regression line of maximum tidal current on tidal range.
  3. Estimate the maximum tidal current on a day when the tidal range is 4.2 m , and comment briefly on how reliable you consider your estimate is likely to be.
  4. It is suggested that the equation found in part (ii) could be used to predict the maximum tidal current on a day when the tidal range is 15 m . Comment briefly on the validity of this suggestion.
OCR S1 2009 January Q1
8 marks Easy -1.2
1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.
NumberProbability
00.7
10.2
20.1
The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).
  1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.
    \(x\)01234
    \(\mathrm { P } ( X = x )\)0.490.280.180.040.01
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2009 January Q2
8 marks Moderate -0.8
2 The table shows the age, \(x\) years, and the mean diameter, \(y \mathrm {~cm}\), of the trunk of each of seven randomly selected trees of a certain species.
Age \(( x\) years \()\)11122028354551
Mean trunk diameter \(( y \mathrm {~cm} )\)12.216.026.439.239.651.360.6
$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$
  1. (a) Use an appropriate formula to show that the gradient of the regression line of \(y\) on \(x\) is 1.13 , correct to 2 decimal places.
    (b) Find the equation of the regression line of \(y\) on \(x\).
  2. Use your equation to estimate the mean trunk diameter of a tree of this species with age
    (a) 30 years,
    (b) 100 years. It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.
  3. Comment on the reliability of each of your two estimates.
OCR S1 2009 January Q3
10 marks Moderate -0.8
3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is \(\frac { 1 } { 8 }\). It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.
  1. Calculate the probability that Erika first sees a woodpecker
    1. on the third day,
    2. after the third day.
    3. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
    4. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.
OCR S1 2009 January Q4
7 marks Moderate -0.3
4 Three tutors each marked the coursework of five students. The marks are given in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)
Tutor 17367604839
Tutor 26250617665
Tutor 34250635471
  1. Calculate Spearman's rank correlation coefficient, \(r _ { \mathrm { s } }\), between the marks for tutors 1 and 2 .
  2. The values of \(r _ { \mathrm { s } }\) for the other pairs of tutors, are as follows. $$\begin{array} { c c } \text { Tutors } 1 \text { and 3: } & r _ { \mathrm { s } } = - 0.9 \\ \text { Tutors } 2 \text { and 3: } & r _ { \mathrm { s } } = 0.3 \end{array}$$ State which two tutors differ most widely in their judgements. Give your reason.
OCR S1 2009 January Q5
8 marks Easy -1.3
5 The stem-and-leaf diagram shows the masses, in grams, of 23 plums, measured correct to the nearest gram.
5567889
61235689
700245678
80
97
9
\(\quad\) Key \(: 6 \mid 2\) means 62
  1. Find the median and interquartile range of these masses.
  2. State one advantage of using the interquartile range rather than the standard deviation as a measure of the variation in these masses.
  3. State one advantage and one disadvantage of using a stem-and-leaf diagram rather than a box-and-whisker plot to represent data.
  4. James wished to calculate the mean and standard deviation of the given data. He first subtracted 5 from each of the digits to the left of the line in the stem-and-leaf diagram, giving the following.
    0567889
    11235689
    200245678
    30
    47
    The mean and standard deviation of the data in this diagram are 18.1 and 9.7 respectively, correct to 1 decimal place. Write down the mean and standard deviation of the data in the original diagram.
OCR S1 2009 January Q6
12 marks Standard +0.3
6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.
  1. (a) How many different arrangements are possible?
    (b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
  2. (a) How many different arrangements are possible?
    (b) Find the probability that questions A and H are next to each other.
    (c) Find the probability that questions B and J are separated by more than four other questions.
OCR S1 2009 January Q7
12 marks Moderate -0.8
7 At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).
  1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
  2. Find
    1. \(\mathrm { P } ( X = 3 )\),
    2. \(\mathrm { P } ( X \geqslant 1 )\).
    3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .
OCR S1 2009 January Q8
7 marks Moderate -0.3
8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
  1. Find the probability that the final score is 4 .
  2. Given that the die is thrown only once, find the probability that the final score is 4 .
  3. Given that the die is thrown twice, find the probability that the final score is 4 .
OCR S1 2011 January Q2
11 marks Moderate -0.8
2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find
  1. \(\mathrm { P } ( X = 3 )\),
  2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
  3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
  4. Find the probability that the total of these two values is 3 .
OCR S1 2011 January Q3
12 marks Moderate -0.8
3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, \(\pounds x\) thousand, and the annual profit, \(\pounds y\) thousand. A summary of the figures for 12 years is as follows. $$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$
  1. Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
  2. Comment briefly on this value in this context.
  3. A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
  4. Calculate the equation of the regression line of \(y\) on \(x\).
  5. Estimate the annual profit during a year when \(\pounds 7400\) was spent on advertising.
OCR S1 2011 January Q4
7 marks Moderate -0.8
4 Jenny and Omar are each allowed two attempts at a high jump.
  1. The probability that Jenny will succeed on her first attempt is 0.6 . If she fails on her first attempt, the probability that she will succeed on her second attempt is 0.7 . Calculate the probability that Jenny will succeed.
  2. The probability that Omar will succeed on his first attempt is \(p\). If he fails on his first attempt, the probability that he will succeed on his second attempt is also \(p\). The probability that he succeeds is 0.51 . Find \(p\). \(530 \%\) of packets of Natural Crunch Crisps contain a free gift. Jan buys 5 packets each week.
OCR S1 2011 January Q6
10 marks Moderate -0.8
6
  1. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.
    1333559
    How many different 7 -digit numbers can be formed using these cards?
  2. The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
    1. How many selections of seven cards are possible?
    2. Find the probability that the seven cards include exactly one card showing the letter A .
OCR S1 2011 January Q7
5 marks Easy -1.2
7 The probability distribution of a discrete random variable, \(X\), is shown below.
\(x\)02
\(\mathrm { P } ( X = x )\)\(a\)\(1 - a\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  2. Show that \(\operatorname { Var } ( X ) = 4 a ( 1 - a )\).