Questions S1 (2020 questions)

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OCR S1 2006 June Q4
7 marks Moderate -0.3
4
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
  3. The random variable \(Z\) has the distribution \(\mathrm { B } ( n , 0.27 )\). Find the smallest value of \(n\) such that \(\mathrm { P } ( Z \geqslant 1 ) > 0.95\).
OCR S1 2006 June Q5
9 marks Moderate -0.8
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 \(1 \frac { 1 } { 4 }\).
  1. Calculate the values of \(p\) and \(q\).
  2. Calculate the standard deviation of \(X\).
OCR S1 2006 June Q6
10 marks Standard +0.3
6 The table shows the total distance travelled, in thousands of miles, and the amount of commission earned, in thousands of pounds, by each of seven sales agents in 2005.
Agent\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Distance travelled18151214162413
Commission earned18451924272223
  1. (a) Calculate Spearman's rank correlation coefficient, \(r _ { s }\), for these data.
    (b) Comment briefly on your value of \(r _ { s }\) with reference to this context.
    (c) After these data were collected, agent \(A\) found that he had made a mistake. He had actually travelled 19000 miles in 2005. State, with a reason, but without further calculation, whether the value of Spearman's rank correlation coefficient will increase, decrease or stay the same. The agents were asked to indicate their level of job satisfaction during 2005. A score of 0 represented no job satisfaction, and a score of 10 represented high job satisfaction. Their scores, \(y\), together with the data for distance travelled, \(x\), are illustrated in the scatter diagram below.
    [diagram]
  2. For this scatter diagram, what can you say about the value of
    (a) Spearman's rank correlation coefficient,
    (b) the product moment correlation coefficient?
OCR S1 2006 June Q7
13 marks Moderate -0.8
7 In a UK government survey in 2000, smokers were asked to estimate the time between their waking and their having the first cigarette of the day. For heavy smokers, the results were as follows.
Time between waking
and first cigarette
1 to 4
minutes
5 to 14
minutes
15 to 29
minutes
30 to 59
minutes
At least 60
minutes
Percentage of smokers312719149
Times are given correct to the nearest minute.
  1. Assuming that 'At least 60 minutes' means 'At least 60 minutes but less than 240 minutes', calculate estimates for the mean and standard deviation of the time between waking and first cigarette for these smokers.
  2. Find an estimate for the interquartile range of the time between waking and first cigarette for these smokers. Give your answer correct to the nearest minute.
  3. The meaning of 'At least 60 minutes' is now changed to 'At least 60 minutes but less than 480 minutes'. Without further calculation, state whether this would cause an increase, a decrease or no change in the estimated value of
    1. the mean,
    2. the standard deviation,
    3. the interquartile range.
OCR S1 2006 June Q8
12 marks Moderate -0.8
8 Henry makes repeated attempts to light his gas fire. He makes the modelling assumption that the probability that the fire will light on any attempt is \(\frac { 1 } { 3 }\). Let \(X\) be the number of attempts at lighting the fire, up to and including the successful attempt.
  1. Name the distribution of \(X\), stating a further modelling assumption needed. In the rest of this question, you should use the distribution named in part (i).
  2. Calculate
    1. \(\mathrm { P } ( X = 4 )\),
    2. \(\mathrm { P } ( X < 4 )\).
    3. State the value of \(\mathrm { E } ( X )\).
    4. Henry has to light the fire once a day, starting on March 1st. Calculate the probability that the first day on which fewer than 4 attempts are needed to light the fire is March 3rd.
OCR S1 2007 June Q1
5 marks Easy -1.8
1 The table shows the probability distribution for a random variable X.
x0123
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)0.10.20.30.4
Calculate \(\mathrm { E } ( \mathrm { X } )\) and \(\operatorname { Var } ( \mathrm { X } )\).
OCR S1 2007 June Q2
5 marks Easy -1.2
2 Two judges each placed skaters from five countries in rank order.
Position1st2nd3rd4th5th
Judge 1UKFranceRussiaPolandCanada
Judge 2RussiaCanadaFranceUKPoland
Calculate Spearman's rank correlation coefficient, \(\mathrm { r } _ { \mathbf { s } ^ { \prime } }\) for the two judges' rankings.
OCR S1 2007 June Q3
4 marks Easy -1.2
3
  1. How many different teams of 7 people can be chosen, without regard to order, from a squad of 15 ?
  2. The squad consists of 6 forwards and 9 defenders. How many different teams containing 3 forwards and 4 defenders can be chosen?
OCR S1 2007 June Q4
8 marks Moderate -0.8
4 A bag contains 6 white discs and 4 blue discs. Discs are removed at random, one at a time, without replacement.
  1. Find the probability that
    1. the second disc is blue, given that the first disc was blue,
    2. the second disc is blue,
    3. the third disc is blue, given that the first disc was blue.
    4. The random variable \(X\) is the number of discs which are removed up to and including the first blue disc. State whether the variable X has a geometric distribution. Explain your answer briefly.
OCR S1 2007 June Q5
10 marks Moderate -0.8
5 The numbers of births, in thousands, to mothers of different ages in England and Wales, in 1991 and 2001 are illustrated by the cumulative frequency curves. Cumulative frequency (000's) \includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-3_949_1338_461_479}
  1. In which of these two years were there more births? How many more births were there in this year?
  2. The following quantities were estimated from the diagram.
    Year
    M edian age
    (years)
    Interquartile
    range (years)
    Proportion of mothers
    giving birth aged below 25
    Proportion of mothers
    giving birth aged 35 or above
    199127.57.3\(33 \%\)\(9 \%\)
    2001\(18 \%\)
    1. Find the values missing from the table.
    2. Did the women who gave birth in 2001 tend to be younger or older or about the same age as the women who gave birth in 1991? Using the table and your values from part (a), give two reasons for your answer.
OCR S1 2007 June Q6
12 marks Moderate -0.3
6 A machine with artificial intelligence is designed to improve its efficiency rating with practice. The table shows the values of the efficiency rating, y , after the machine has carried out its task various numbers of times, \(x\)
x0123471330
y0481011121314
$$\left[ n = 8 , \Sigma x = 60 , \Sigma y = 72 , \Sigma x ^ { 2 } = 1148 , \Sigma y ^ { 2 } = 810 , \Sigma x y = 767 . \right]$$ These data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-4_769_1328_760_411}
  1. (a) Calculate the value of r , the product moment correlation coefficient.
    (b) Without calculation, state with a reason the value of \(\mathrm { r } _ { \mathrm { s } ^ { \prime } }\) Spearman's rank correlation coefficient.
  2. A researcher suggests that the data for \(\mathrm { x } = 0\) and \(\mathrm { x } = 1\) should be ignored. Without cal culation, state with a reason what effect this would have on the value of
    (a) \(r\),
    (b) \(r _ { s }\).
  3. Use the diagram to estimate the value of y when \(\mathrm { x } = 29\).
  4. Jack finds the equation of the regression line of y on xf for all the data, and uses it to estimate the value of \(y\) when \(x = 29\). Without calculation, state with a reason whether this estimate or the one found in part (iii) will be the more reliable.
OCR S1 2007 June Q7
9 marks Moderate -0.3
7 On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by X .
  1. State two conditions needed for X to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
  2. Find \(\mathrm { P } ( \mathrm { X } \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
  3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
  4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
OCR S1 2007 June Q8
8 marks Moderate -0.8
8
  1. A biased coin is thrown twice. The probability that it shows heads both times is 0.04 . Find the probability that it shows tails both times.
  2. A nother coin is biased so that the probability that it shows heads on any throw is p . The probability that the coin shows heads exactly once in two throws is 0.42 . Find the two possible values of p.
OCR S1 2007 June Q9
11 marks Standard +0.3
9
  1. A random variable \(X\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find
    1. \(\mathrm { E } ( \mathrm { X } )\),
    2. \(\mathrm { P } ( \mathrm { X } = 4 )\),
    3. \(P ( X > 4 )\).
    4. A random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\), and \(q = 1 - p\).
      (a) Show that \(P ( Y\) is odd \() = p + q ^ { 2 } p + q ^ { 4 } p + \ldots\).
      (b) Use the formula for the sum to infinity of a geometric progression to show that $$P ( Y \text { is odd } ) = \frac { 1 } { 1 + q }$$ {}
      7
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.