Questions FS1 AS (68 questions)

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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]
OCR FS1 AS 2021 June Q2
8 marks Standard +0.3
The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(P(W = w)\)0.250.36\(x\)\(x^2\)
  1. Show that \(\text{Var}(W) = 0.8571\). [7]
  2. Find \(\text{Var}(3W + 6)\). [1]
OCR FS1 AS 2021 June Q3
5 marks Standard +0.8
In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(E(T) = 5.625\) and the standard deviation of \(T\) is \(1.875\). Find the values of the parameters of the distribution. [5]
OCR FS1 AS 2021 June Q4
9 marks Challenging +1.2
The table shows the results of a random sample drawn from a population which is thought to have the distribution \(U(20)\).
Range\(1 \leq x \leq 8\)\(9 \leq x \leq 12\)\(13 \leq x \leq 20\)
Observed frequency12\(y\)\(28 - y\)
Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5\%\) significance level. [9]
OCR FS1 AS 2017 Specimen Q1
5 marks Moderate -0.3
Two music critics, \(P\) and \(Q\), give scores to seven concerts as follows.
Concert1234567
Score by critic \(P\)1211613171614
Score by critic \(Q\)913814181620
  1. Calculate Spearman's rank correlation coefficient, \(r_s\), for these scores. [4]
  2. Without carrying out a hypothesis test, state what your answer tells you about the views of the two critics. [1]
OCR FS1 AS 2017 Specimen Q2
7 marks Standard +0.3
The probability distribution of a discrete random variable \(W\) is given in the table.
\(w\)0123
\(\mathrm{P}(W = w)\)0.190.18\(x\)\(y\)
Given that \(\mathrm{E}(W) = 1.61\), find the value of \(\text{Var}(3W + 2)\). [7]
OCR FS1 AS 2017 Specimen Q3
8 marks Standard +0.3
Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below.
BlackNon-black
Men6971
Women3055
Test at the 5\% significance level whether Carl's belief is justified. [8]
OCR FS1 AS 2017 Specimen Q4
6 marks Standard +0.8
  1. Four men and four women stand in a random order in a straight line. Determine the probability that no one is standing next to a person of the same gender. [3]
  2. \(x\) men, including Mr Adam, and \(x\) women, including Mrs Adam, are arranged at random in a straight line. Show that the probability that Mr Adam is standing next to Mrs Adam is \(\frac{1}{x}\). [3]
OCR FS1 AS 2017 Specimen Q5
7 marks Standard +0.3
  1. The random variable \(X\) has the distribution \(\text{Geo}(0.6)\).
    1. Find \(\mathrm{P}(X \geq 8)\). [2]
    2. Find the value of \(\mathrm{E}(X)\). [1]
    3. Find the value of \(\text{Var}(X)\). [1]
  2. The random variable \(Y\) has the distribution \(\text{Geo}(p)\). It is given that \(\mathrm{P}(Y < 4) = 0.986\) correct to 3 significant figures. Use an algebraic method to find the value of \(p\). [3]
OCR FS1 AS 2017 Specimen Q6
13 marks Moderate -0.3
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
  1. State these two assumptions. [2]
  2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\text{Po}(0.8)\).
    1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
    2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
  2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
  3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\text{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]
OCR FS1 AS 2017 Specimen Q7
4 marks Standard +0.3
The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.
\(x\)012
Observed frequency\(N - 1\)\(N - 1\)\(N + 2\)
The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]
OCR FS1 AS 2017 Specimen Q8
10 marks Standard +0.3
The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.
\(x\)9.39.79.79.79.910.210.511.010.610.6
\(y\)480501540552547622655701712708
source: www.tylervigen.com
  1. Find the equation of the regression line of \(y\) on \(x\). [2]
You are given that the product moment correlation coefficient is 0.959.
  1. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. [1]
It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.
  1. Assume that the data is a random sample of all years.
    1. Carry out the test at the 10\% significance level. [6]
    2. Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. [1]