Questions — OCR FS1 AS (36 questions)

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OCR FS1 AS 2017 December Q1
8 marks Moderate -0.3
1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that \(5 \%\) of letters receive a favourable reply.
  1. Bill sends a letter to each of 40 potential sponsors. Assuming that the number \(N\) of favourable responses can be modelled by a binomial distribution, find the mean and variance of \(N\).
  2. Gill sends one letter at a time to potential sponsors. \(L\) is the number of letters she sends, up to and including the first letter that receives a favourable response.
    1. State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
    2. Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a \(90 \%\) chance of receiving at least one favourable reply.
OCR FS1 AS 2017 December Q2
7 marks Moderate -0.3
2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.
  1. All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.
  2. 4 cards are chosen at random. Find the probability that at least three consonants ( \(\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }\) ) are on the cards chosen.
OCR FS1 AS 2017 December Q3
7 marks Standard +0.3
3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.
  1. Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
  2. Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of \(t\) minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
  3. Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
  4. Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.
OCR FS1 AS 2017 December Q4
10 marks Standard +0.3
4 The discrete random variable \(X\) has the distribution \(\mathrm { U } ( n )\).
  1. Use the results \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) and \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) to show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\). It is given that \(\mathrm { E } ( X ) = 13\).
  2. Find the value of \(n\).
  3. Find \(\mathrm { P } ( X < 7.5 )\). It is given that \(\mathrm { E } ( a X + b ) = 10\) and \(\operatorname { Var } ( a X + b ) = 117\), where \(a\) and \(b\) are positive.
  4. Calculate the value of \(a\) and the value of \(b\).
OCR FS1 AS 2017 December Q5
8 marks Moderate -0.5
5 A shop manager recorded the maximum daytime temperature \(T ^ { \circ } \mathrm { C }\) and the number \(C\) of ice creams sold on 9 summer days. The results are given in the table and illustrated in the scatter diagram.
\(T\)172125262727293030
\(C\)211620383237353942
\includegraphics[max width=\textwidth, alt={}]{64d7ed6d-fadd-4c59-afb0-97d1788ba369-3_661_1189_1320_431}
$$n = 9 , \Sigma t = 232 , \Sigma c = 280 , \Sigma t ^ { 2 } = 6130 , \Sigma c ^ { 2 } = 9444 , \Sigma t c = 7489$$
  1. State, with a reason, whether one of the variables \(C\) or \(T\) is likely to be dependent upon the other.
  2. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
  3. State with a reason what the value of \(r\) would have been if the temperature had been measured in \({ } ^ { \circ } \mathrm { F }\) rather than \({ } ^ { \circ } \mathrm { C }\).
  4. Calculate the equation of the least squares regression line of \(c\) on \(t\).
  5. The regression line is drawn on the copy of the scatter diagram in the Printed Answer Booklet. Use this diagram to explain what is meant by "least squares".
OCR FS1 AS 2017 December Q6
9 marks Standard +0.3
6 Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.
  1. Arlosh calculates Spearman's rank correlation coefficient \(r _ { s }\) for the critics' ratings. He calculates that \(\Sigma d ^ { 2 } = 72\). Show that this value must be incorrect.
  2. Arlosh checks his working with Sarah, whose answer \(r _ { s } = \frac { 29 } { 35 }\) is correct. Find the correct value of \(\Sigma d ^ { 2 }\).
  3. Carry out an appropriate two-tailed significance test of the value of \(r _ { s }\) at the \(5 \%\) significance level, stating your hypotheses clearly. Each critic gives a score out of 100 to each film. Desi uses these scores to calculate Pearson's product-moment correlation coefficient. She carries out a two-tailed significance test of this value at the \(5 \%\) significance level.
  4. Explain with a reason whether you would expect the conclusion of Desi's test to be the same as the result of the test in part (iii).
OCR FS1 AS 2017 December Q7
11 marks Standard +0.3
7 Josh is investigating whether sticking pins into a map at random, while blindfolded, provides a random sample of regions of the map. Josh divides the map into 49 squares of equal size and asks each of 98 friends to stick a pin into the map at random, while blindfolded. He then notes the number of pins in each square. To analyse the results he groups the squares as shown in the diagram.
DDDDDDD
DCCCCCD
DCBBBCD
DCBABCD
DCBBBCD
DCCCCCD
DDDDDDD
The results are summarised in the table.
RegionABCD
Number of squares181624
Number of pins6213338
  1. Test at the 10\% significance level whether the use of pins in this way provides a random sample of regions of the map.
  2. What can be deduced from considering the different contributions to the test statistic? \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR FS1 AS 2018 March Q1
7 marks Easy -1.2
1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by \(X\).
  1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X \sim \operatorname { Geo } ( 0.4 )\).
  2. Find \(\mathrm { P } ( X < 6 )\).
  3. Find \(\mathrm { E } ( X )\).
  4. Find \(\operatorname { Var } ( X )\).
OCR FS1 AS 2018 March Q2
6 marks Standard +0.3
2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).
  1. It is given that \(W\) has a Poisson distribution.
    1. Write down the standard deviation of \(W\).
    2. Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
    3. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).
OCR FS1 AS 2018 March Q3
8 marks Standard +0.8
3 A pack of 40 cards consists of 10 cards in each of four colours: red, yellow, blue and green. The pack is dealt at random into four "hands", each of 10 cards. The hands are labelled North, South, East and West.
  1. Find the probability that West has exactly 3 red cards.
  2. Find the probability that West has exactly 3 red cards, given that East and West have between them a total of exactly 5 red cards.
  3. South has 5 red cards and 5 blue cards. These cards are placed in a row in a random order. Find the probability that the colour of each card is different from the colour of the preceding card.
OCR FS1 AS 2018 March Q4
9 marks Moderate -0.3
4 A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table.
Score, \(N\)12345
Probability0.30.20.2\(x\)\(y\)
It is known that \(\mathrm { E } ( N ) = 2.55\).
  1. Find \(\operatorname { Var } ( N )\).
  2. Find \(\mathrm { E } ( 3 N + 2 )\).
  3. Find \(\operatorname { Var } ( 3 N + 2 )\).
OCR FS1 AS 2018 March Q5
4 marks Easy -1.8
5 The speed \(v \mathrm {~ms} ^ { - 1 }\) of a car at time \(t\) seconds after it starts to accelerate was measured at 1 -second intervals. The results are shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5843350-52f9-4fed-adf4-86ceb958033f-3_661_1186_1078_443}
  1. State whether \(t\) or \(v\) or neither is a controlled variable. The value of the product moment correlation coefficient \(r\) for the data is 0.987 correct to 3 significant figures.
  2. The speed of the car is converted to miles per hour and the time to minutes. State the value of \(r\) for the converted data.
  3. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
  4. What information does \(r\) give about the data that is not given by \(r _ { s }\) ?
OCR FS1 AS 2018 March Q6
7 marks Challenging +1.2
6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]
OCR FS1 AS 2018 March Q7
11 marks Standard +0.3
7 The numbers of students taking A levels in three subjects at a school were classified by the year in which they entered the school as follows.
\cline { 2 - 5 } \multicolumn{1}{c|}{}SubjectMathematicsEnglishPhysics
\multirow{3}{*}{
Year of
Entry
}		Year 717167
\cline { 2 - 5 }Year 121325
The Head of the school carries out a significance test at the \(10 \%\) level to test whether subjects taken are independent of year of entry.
  1. Show that in carrying out the test it is necessary to combine columns.
  2. Suggest a reason why it is more sensible to combine the columns for Mathematics and Physics than the columns for Physics and English.
  3. Carry out the test.
  4. State which cell gives the largest contribution to the test statistic.
  5. Interpret your answer to part (iv).
OCR FS1 AS 2018 March Q8
8 marks Challenging +1.2
8 In a competition, entrants have to give ranks from 1 to 7 to each of seven resorts. The correct ranks for the resorts are decided by an expert.
  1. One competitor chooses his ranks randomly. By considering all the possible rankings, find the probability that the value of Spearman's rank correlation coefficient \(r _ { s }\) between the competitor's ranks and the expert's ranks is at least \(\frac { 27 } { 28 }\).
  2. Another competitor ranks the seven resorts. A significance test is carried out to test whether there is evidence that this competitor is merely guessing the rank order of the seven resorts. The critical region is \(r _ { s } \geqslant \frac { 27 } { 28 }\). State the significance level of the test. \section*{END OF QUESTION PAPER}
OCR FS1 AS 2021 June Q1
8 marks Moderate -0.8
1 The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(\mathrm { P } ( W = w )\)0.250.36\(x\)\(x ^ { 2 }\)
  1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
  2. Find \(\operatorname { Var } ( 3 W + 6 )\).
OCR FS1 AS 2021 June Q2
6 marks Moderate -0.8
2 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).
  1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
  2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
  3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4 . Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \Sigma x = 3600 , \Sigma x ^ { 2 } = 660500 , \Sigma y = 1440 , \Sigma y ^ { 2 } = 105280 , \Sigma x y = 260990$$
    1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
    2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
    3. Assume now that the assumption in part (b) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
    4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
    5. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
    6. State an assumption needed to be able to carry out a significance test on the value of \(r\).
    7. Assume now that the assumption in part (b) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
    8. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\mathrm { ELO } \text { rating } = 8 \times \mathrm { BCF } \text { rating } + 650 .$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion. An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.
      QuestionAnswerMarksAOGuidance
      1(a)\(\begin{aligned}0.25 + 0.36 + x + x ^ { 2 } = 1
      x ^ { 2 } + x - 0.39 = 0
      x = 0.3 \text { (or } - 1.3 \text { ) }
      x \text { cannot be negative }
      \mathrm { E } ( W ) = 2.23
      \mathrm { E } \left( W ^ { 2 } \right) = \Sigma w ^ { 2 } \mathrm { p } ( w ) \quad [ = 5.83 ]
      \text { Subtract } [ \mathrm { E } ( W ) ] ^ { 2 } \text { to get } \mathbf { 0 . 8 5 7 1 } \end{aligned}\)\(\begin{gathered} \text { M1 }
      \text { A1 }
      \text { A1 }
      \text { B1ft }
      \text { B1 }
      \text { M1 }
      \text { A1 }
      { [ 7 ] } \end{gathered}\)
      3.1a
      1.1b
      1.1b
      2.3
      1.1b
      1.1
      2.1
      Equation using \(\Sigma p = 1\)
      Correct simplified quadratic Correctly obtain \(x = 0.3\)
      Explicitly reject other solution
      2.23 or exact equivalent only Use \(\Sigma w ^ { 2 } \mathrm { p } ( w )\)
      Correctly obtain given answer, www
      Can be implied
      Method needed ft on their quadratic Allow for \(\mathrm { E } ( W ) ^ { 2 } = 4.9729\)
      Need 2.23 or 4.9729 and 5.83 or full numerical \(\Sigma w ^ { 2 } \mathrm { p } ( w )\)
      1(b)\(9 \times 0.8571 = 7.7139\)
      B1
      [1]
      		1.1bAllow 7.71 or 7.714
      2(a)Flaws must occur at constant average rate (uniform rate)
      B1
      [1]
      		1.2
      Context (e.g. "flaws") needed
      Extra answers, e.g. "singly": B0
      		Not "constant rate" or "average constant rate".
      2(b)\(\operatorname { Po(2.1)~or~ } e ^ { - \lambda } \frac { \lambda ^ { 3 } } { 3 ! }\)
      M1
      A1
      [2]
      1.1
      1.1b
      		Po(2.1) stated or implied, or formula with \(\lambda = 2.1\) stated Awrt 0.189
      2(c)
      Po(3)
      \(1 - \mathrm { P } ( \leq 3 )\)
      M1
      M1
      A1
      [3]
      1.1
      1.1
      1.1b
      \(\operatorname { Po } ( 2 \times 0.7 + 1.6 )\) stated or implied
      Allow \(1 - \mathrm { P } ( \leq 4 ) = 0.1847\), or from wrong \(\lambda\)
      Awrt 0.353
      Or all combinations \(\leq 3\)
      \(1 -\) above, not just \(= 3\)
      QuestionAnswerMarksAOGuidance
      3(a)0.4(00)
      B2
      [2]
      1.1
      1.1b
      		SC: if B0, give SC B1 for two of \(S _ { x x } = 12500 , S _ { y y } = 1600 , S _ { x y } = 1790\) and \(S _ { x y } / \sqrt { } \left( S _ { x x } S _ { y y } \right)\)Also allow SC B1 for equivalent methods using Covariance \SDs
      3(b)Data needs to have a bivariate normal distribution
      B1
      [1]
      		1.2Needs "bivariate normal" or clear equivalent. Not just "both normally distributed"Allow "scatter diagram forms ellipse"
      3(c)
      \(\mathrm { H } _ { 0 }\) : higher maths scores are not associated with higher BCF grading; \(\mathrm { H } _ { 1 }\) : positively associated
      CV 0.3783
      \(0.400 > 0.3783\) so reject \(\mathrm { H } _ { 0 }\)
      Significant evidence that higher maths scores are associated with higher BCF grading
      B1
      B1
      M1ft
      A1ft
      [4]
      2.5
      1.1b
      2.2b
      3.5a
      Needs context and clearly onetailed \(O R \rho\) used and defined Not "evidence that ..."
      Allow 0.378
      Reject/do not reject \(\mathrm { H } _ { 0 }\)
      Contextualised, not too definite Needn't say "positive" if \(\mathrm { H } _ { 1 } \mathrm { OK }\)
      SC 2-tail: B0; 0.4438, or 0.3783 B1; then M1A0
      \(\mathrm { H } _ { 0 } : \rho = 0 , \mathrm { H } _ { 1 } : \rho > 0\) where \(\rho\) is population pmcc (not \(r\) )
      FT on their \(r\), but not CV
      Not "scores are associated
      ...". FT on their \(r\) only
      3(d)It makes no difference as this is a linear transformation
      B1
      [1]
      		2.2aNeed both "unchanged" oe and reason, need "linear" or exact equivalent"oe" includes "their 0.4"
      4(a)Neither
      B1
      [1]
      		2.5OENot "neither is independent of the other"
      4(b)\(c = 2.848 - 0.1567 m\)
      B1
      B1
      B1
      [3]
      1.1
      1.1
      1.1
      Correct \(a\), awrt 2.85
      Correct \(b\), awrt 0.157
      Letters correct from correct method
      (If both wrongly rounded, e.g. \(c = 2.84 - 0.156 m\), give B2)
      \(\mathrm { SC } : m\) on \(c\) :
      \(m = 15.65 - 4.832 c\) : B2
      \(y = 15.65 - 4.832 x\) : B1
      \(c = 15.65 - 4.832 m : \mathrm { B } 1\)
      If B0B0, give B1 for correct letters from valid working
      QuestionAnswerMarksAOGuidance
      4(c)\(a\) unchanged, \(b\) multiplied by 2.2 (allow " \(a\) unchanged, \(b\) increases", etc)B1 [1]2.2aoe, e.g. \(c = 2.848 - 0.345 m\); \(m = 7.114 - 2.196 c\)SC: \(m\) on \(c\) in (b): Both divided by 2.2 B1
      4(d)
      Draw approximate line of best fit
      Draw at least one vertical from line to point
      Say that "Best fit" line minimises the sum of squares of these distances
      M1
      M1
      A1
      [3]
      1.1
      2.4
      2.4
      		Needs M2 and "minimises" and "sums of squares" oe
      SC: Horizontal(s):
      full marks (indept of (b))
OCR FS1 AS 2021 June Q1
6 marks Standard +0.3
1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\mathrm { Po } ( 120 )\).
  1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
  2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
  3. State a necessary assumption for the validity of your calculation in part (b).
OCR FS1 AS 2021 June Q2
7 marks Standard +0.3
2 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.
  1. Find the probability that all the men are next to each other.
  2. Find the probability that no two men are next to one another.
OCR FS1 AS 2021 June Q3
5 marks Moderate -0.3
3 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.
  1. For all sixteen candidates, the value of the product moment correlation coefficient \(r\) for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the \(5 \%\) significance level, of association between the marks on the two papers.
  2. A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by \(n = 10 , \Sigma x = 750 , \Sigma y = 690 , \Sigma x ^ { 2 } = 57690 , \Sigma y ^ { 2 } = 49676 , \Sigma x y = 50829\).
    1. Calculate the value of \(r\) for these 10 candidates.
    2. What do the two values of \(r\), in parts (a) and (b)(i), tell you about the scores of the sixteen candidates? A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by \(p\).
      1. Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of \(p\), for the variance of the number of green beads Sasha selects. Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the \(X\) th bead that Freda selects.
      2. Assume that \(p = 0.3\). Find
        1. \(\mathrm { P } ( X \geqslant 5 )\),
        2. \(\operatorname { Var } ( X )\).
    3. In fact, on the basis of a large number of observations of \(X\), it is found that \(\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )\). Estimate the value of \(p\).
OCR FS1 AS 2021 June Q1
5 marks Moderate -0.3
1 Five observations of bivariate data \(( x , y )\) are given in the table.
\(x\)781264
\(y\)201671723
  1. Find the value of Pearson's product-moment correlation coefficient.
  2. State what your answer to part (a) tells you about a scatter diagram representing the data.
  3. A new variable \(a\) is defined by \(a = 3 x + 4\). Dee says "The value of Pearson's product-moment correlation coefficient between \(a\) and \(y\) will not be the same as the answer to part (a)." State with a reason whether you agree with Dee. An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods. The profit in the first year for each account is \(p \%\) and the profit in the second year for each account is \(q \%\). The results are shown in the table and in the scatter diagram.
    AccountABCDEFGH
    \(p\)1.62.12.42.72.83.35.28.4
    \(q\)1.62.32.22.23.12.97.64.8
    \(n = 8 \quad \Sigma p = 28.5 \quad \Sigma q = 26.7 \quad \Sigma p ^ { 2 } = 136.35 \quad \Sigma q ^ { 2 } = 116.35 \quad \Sigma p q = 116.70\) \includegraphics[max width=\textwidth, alt={}, center]{4c7546b9-03ee-47a1-915f-41e2b4ca19c0-03_762_1248_906_260}
    1. State which, if either, of the variables \(p\) and \(q\) is independent.
    2. Calculate the equation of the regression line of \(q\) on \(p\).
      1. Use the regression line to estimate the value of \(q\) for an investment account for which \(p = 2.5\).
      2. Give two reasons why this estimate could be considered reliable.
    4. Comment on the reliability of using the regression line to predict the value of \(q\) when \(p = 7.0\).
OCR FS1 AS 2021 June Q3
12 marks Standard +0.3
3 At a cinema there are three film sessions each Saturday, "early", "middle" and "late". The numbers of the audience, in different age groups, at the three showings on a randomly chosen Saturday are given in Table 1. \begin{table}[h] \end{table}
QuestionSolutionMarksAOsGuidance
1(a)-0.954 BCB2 [2]1.1 1.1SC: If B0, give B1 if two of 7.04, 29.0[4], -13.6[4] (or 35.2, 145[.2], -68.2) seen
1(b)Points lie close to a straight line Line has negative gradientB1 B1 [2]2.2b 1.1Must refer to line, not just "negative correlation"
1(c)No, it will be the same as \(x \rightarrow a\) is a linear transformationB1 [1]2.2aOE. Either "same" with correct reason, or "disagree" with correct reason. Allow any clear valid technical term
2(a)NeitherB1 [1]1.2
2(b)\(q = 1.13 + 0.620 p\)B1B1 B1 [3]1.1,1.1 1.10.62(0) correct; both numbers correct Fully correct answer including letters
2(c)(i)2.68B1ft [1]1.1awrt 2.68, ft on their (b) if letters correct
2(c)(ii)2.5 is within data range, and points (here) are close to line/well correlatedB1 B1 [2]2.2b 2.2bAt least one reason, allow "no because points not close to line" Full argument, two reasons needed
2(d)
Not much data here/points scattered/ possible outliers
So not very reliable
M1 A1 [2]2.3 1.1Reason for not very reliable (not "extrapolation") Full argument and conclusion, not too assertive (not wholly unreliable!)
3(a)Expected frequency for Middle/25 to 60 is 4.4 which is < 5 so must combine cellsB1*ft depB1 [2]2.4 3.5bCorrectly obtain this \(F _ { E }\), ft on addition errors " < 5" explicit and correct deduction
3(b)
EarlyMiddleLate
29.423.131.5
26.620.928.5
EarlyMiddleLate
0.99180.41602.2937
1.09620.45982.5351
B11.1
Both, allow 28.4 for 28.5
awrt 2.29, but allow 2.3 In range [2.53, 2.54]
QuestionSolutionMarksAOsGuidance
3(c)
\(\mathrm { H } _ { 0 }\) : no association between session and age group. \(\mathrm { H } _ { 1 }\) : some association
\(\Sigma X ^ { 2 } = 7.793\)
\(v = 2 , \chi ^ { 2 } ( 2 ) _ { \text {crit } } = 5.991\)
Reject \(\mathrm { H } _ { 0 }\).
Significant evidence of association between session attended and age group.
B1
B1
B1
M1ft
A1ft [5]
1.1
1.1
1.1
1.1
2.2b
Both. Allow "independent" etc
Correct value of \(X ^ { 2 }\), awrt 7.79 (allow even if wrong in (b))
Correct CV and comparison
Correct first conclusion, FT on their TS only
Contextualised, not too assertive
3(d)The two biggest contributions to \(\chi ^ { 2 }\) are both for the late session ... ... when the proportion of younger people is higher, and of older people is lower, than the null hypothesis would suggest.
M1ft
A1ft
[2]
1.1
2.4
Refer to biggest contribution(s), FT on their answers to (b), needs "reject \(\mathrm { H } _ { 0 }\) "
Full answer, referring to at least one cell (ignore comments on next highest cells)
\multirow[t]{2}{*}{4}\multirow{2}{*}{}\multirow{2}{*}{OR:}
\(\frac { { } ^ { 2 m } C _ { 2 } \times m } { { } ^ { 3 m } C _ { 3 } }\)
\(= \frac { 2 m ( 2 m - 1 ) } { 2 } \times m \div \frac { 3 m ( 3 m - 1 ) ( 3 m - 2 ) } { 6 }\)
\(= \frac { 2 m ( 2 m - 1 ) } { ( 3 m - 1 ) ( 3 m - 2 ) }\) \(\frac { 2 m ( 2 m - 1 ) } { ( 3 m - 1 ) ( 3 m - 2 ) } = \frac { 28 } { 55 }\)
\(\Rightarrow 16 m ^ { 2 } - 71 m + 28 = 0\)
\(m = 4\) BC
Reject \(m = \frac { 7 } { 16 }\) as \(m\) is an integer
M1
M1
A1
M1
A1
M1
A1
[7]
3.1b
3.1b
2.1
3.1a
2.1
1.1
3.2a
Use \({ } ^ { 2 m } C _ { 2 }\) and \(m\)
Divide by \({ } ^ { 3 m } C _ { 3 }\)
Correct expression in terms of \(m\) (allow with \(m\) not cancelled yet)
Equate to \(\frac { 28 } { 55 }\) \simplify to three-term quadratic
Correct simplified quadratic, or (quadratic) \(\times m , = 0\), aef Solve to get both 4 and \(\frac { 7 } { 16 }\)
Explicitly reject \(m = \frac { 7 } { 16 }\)
\(\frac { 2 m ( 2 m - 1 ) \times m \times 3 ! } { 3 m ( 3 m - 1 ) ( 3 m - 2 ) \times 2 }\) then as above
Multiplication method can get full marks, but if no 3 or 3 !, max
M1M0A0 M1A0M0A0
OCR FS1 AS 2021 June Q1
8 marks Moderate -0.3
1 Every time a spinner is spun, the probability that it shows the number 4 is 0.2 , independently of all other spins.
  1. A pupil spins the spinner repeatedly until it shows the number 4 . Find the mean of the number of spins required.
  2. Calculate the probability that the number of spins required is between 3 and 10 inclusive.
  3. Each pupil in a class of 30 spins the spinner until it shows the number 4 . Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\).
OCR FS1 AS 2021 June Q2
9 marks Moderate -0.5
2 After a holiday organised for a group, the company organising the holiday obtained scores out of 10 for six different aspects of the holiday. The company obtained responses from 100 couples and 100 single travellers. The total scores for each of the aspects are given in the following table.
QuestionAnswerMarkAOGuidance
1(a)\(\frac { 1 } { 0.2 } = 5\)M1 A1 [2]3.3 1.1Geometric distribution soi 5 (or \(5.00 \ldots\) ) only
1(b)\(0.8 ^ { 2 } - 0.8 ^ { 10 }\) \(= \mathbf { 0 . 5 3 3 } \quad ( 0.5326258 \ldots )\)M1 A1 [2]1.1 3.4
Allow for powers 2, 3, 4 and 9, 10, 11 .
Awrt 0.533, www. [5201424/976562]
Or \(0.2 \left( 0.8 ^ { 2 } + \ldots . + 0.8 ^ { 9 } \right) , \pm 1\) term at either end [0.506, 0.378, 0.275, 0.405, 0.302, 0.554, 0.426, 0.324]
1(c)
\(\mathrm { P } ( \geq 10 ) = 0.8 ^ { 9 }\)
\(= 0.1342 \ldots\)
B(30, 0.1342...)
Variance \(= n p q\) = 3.486...
M1
A1
M1
A1ft [4]
3.1b
1.1
3.1b
1.1
Or \(0.8 ^ { 10 }\). Can be implied by correct \(p\)
[0.10737... is M1A0 here]
Stated or implied, their \(0.8 ^ { 9 }\) or \(0.8 ^ { 10 }\)
In range [3.48, 3.49]
SC: 0.134(2) oe not properly shown: B2 for correct final answer.
SC: 2.875 from \(0.8 ^ { 10 }\) : M1A0M1A1ft
QuestionAnswerMarkAOGuidance
2(a)Test is for rankings/rankings arbitrary/not bivariate normal etcB1 [1]2.4OE
2(b)
\(\mathrm { H } _ { 0 } : \rho _ { s } = 0 , \mathrm { H } _ { 1 } : \rho _ { s } > 0\), where \(\rho _ { s }\) is the population rank correlation coefficient
Ranks 543612
512643
\(\Sigma d ^ { 2 } = 20\)
\(r _ { s } = 1 - \frac { 6 \times 20 } { 6 \times 35 }\)
\(= 3 / 7\) or \(0.42857 \ldots\)
<0.9429
B1
B1
M1
A1
B1
1.1
1.1
1.1
1.1
1.1
Allow \(\rho _ { s }\) not defined; allow \(\rho\).
Allow: \(\mathrm { H } _ { 0 }\) : no association between rankings.
\(\mathrm { H } _ { 1 }\) : positive association (but not \(\mathrm { H } _ { 1 }\) : association)
Do not reject \(\mathrm { H } _ { 0 }\)
Insufficient evidence of association between ranking given by the two categories
M1ft
A1ft
[7]
1.1
2.2b
FT on their \(\Sigma d ^ { 2 }\) only
2(c)Not dependent on any distributional assumptions
B1
[1]
1.2Oe (cf. Specification, 5.08f)
QuestionAnswerMarkAOGuidance
3(a)Failures occur to no fixed pattern/are not predictableB1 [1]1.1OE. NOT "independent"
3(b)Failures occur independently of one another and at constant average rate
B1
B1
[2]
1.1
1.1
Not recoverable from (a) if independence not restated here; must be contextualised
Ignore "singly". Allow "uniform" rate, not "constant rate" or "constant probability"; must be contextualised
3(c)
Variance (1.6384) \(\approx\) mean
So suggests that it is likely to be well modelled
M1
A1
[2]
1.1
3.5a
Compare variance (or SD). Allow square/square-root confusion
Correct comparison and conclusion, 1.64 or better seen
3(d)\(\mathrm { e } ^ { - 1.61 }\)
B1
[1]
3.4Exact needed, allow even if \(0 !\) or \(1.61 ^ { 0 }\) or both left in
3(e)
1\(\geq 2\)
0.3220.478
B1
B1
[2]
3.4
1.1
One correct e.g. 0.3218
Other correct e.g. 0.4783
3(f)\(\mathrm { P } ( F = 1 )\) will be smaller as single failures are less likely
B1*
depB1
[2]
3.5c
3.3
OE. Partial answer: B1
OCR FS1 AS 2021 June Q1
8 marks Moderate -0.8
A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is \(0.1\). The first weekday in September on which he receives a delivery of books to review is the \(X\)th weekday of September.
  1. State an assumption needed for \(X\) to be well modelled by a geometric distribution. [1]
  2. Find \(P(X = 11)\). [2]
  3. Find \(P(X \leq 8)\). [2]
  4. Find \(\text{Var}(X)\). [2]
  5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review. [1]