Questions FS1 AS (62 questions)

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Edexcel FS1 AS 2024 June Q2
  1. A manager keeps a record of accidents in a canteen.
Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.
  1. Give a reason why a Poisson distribution could be a suitable model in this situation.
  2. Assuming that a Poisson model is suitable, find the probability of
    1. at least 3 accidents in the next month,
    2. no more than 10 accidents in a 3-month period,
    3. at least 2 months with no accidents in an 8-month period. One day, two members of staff bump into each other in the canteen and each report the accident to the manager. The canteen manager is unsure whether to record this as one or two accidents. Given that the manager still wants to model the number of accidents per month with a Poisson distribution,
  3. state
    • a property of the Poisson distribution that the manager should consider when deciding how to record this situation
    • whether the manager should record this as one or two accidents
    The manager introduces some new procedures to try and reduce the average number of accidents per month. During the following 12 months the total number of accidents is 22 The manager claims that the accident rate has been reduced.
  4. Use a \(5 \%\) level of significance to carry out a suitable test to assess the manager's claim.
    You should state your hypotheses clearly and the \(p\)-value used in your test.
Edexcel FS1 AS 2024 June Q3
  1. The discrete random variable \(X\) has probability distribution,
\(x\)- 10137
\(\mathrm { P } ( X = x )\)\(p\)\(r\)\(p\)0.3\(r\)
where \(p\) and \(r\) are probabilities.
Given that \(\mathrm { E } ( X ) = 1.95\)
find the exact value of \(\mathrm { E } ( \sqrt { X + 1 } )\) giving your answer in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
(6)
Edexcel FS1 AS 2024 June Q4
  1. Robin shoots 8 arrows at a target each day for 100 days.
The number of times he hits the target each day is summarised in the table below.
Number of hits012345678
Frequency1103034174202
Misha believes that these data can be modelled by a binomial distribution.
  1. State, in context, two assumptions that are implied by the use of this model.
  2. Find an estimate for the proportion of arrows Robin shoots that hit the target. Misha calculates expected frequencies, to 2 decimal places, as follows.
    Number of hits012345678
    Expected frequency2.8112.67\(r\)28.0519.73\(s\)2.500.400.03
  3. Find the value of \(r\) and the value of \(s\) Misha correctly used a suitable test to assess her belief.
    1. Explain why she used a test with 3 degrees of freedom.
    2. Complete the test using a \(5 \%\) level of significance. You should clearly state your hypotheses, test statistic, critical value and conclusion.
Edexcel FS1 AS Specimen Q1
  1. A university foreign language department carried out a survey of prospective students to find out which of three languages they were most interested in studying.
A random sample of 150 prospective students gave the following results.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Language
\cline { 3 - 5 } \multicolumn{2}{c|}{}FrenchSpanishM andarin
\multirow{2}{*}{Gender}M ale232220
\cline { 2 - 5 }Female383215
A test is carried out at the \(1 \%\) level of significance to determine whether or not there is an association between gender and choice of language.
  1. State the null hypothesis for this test.
  2. Show that the expected frequency for females choosing Spanish is 30.6
  3. Calculate the test statistic for this test, stating the expected frequencies you have used.
  4. State whether or not the null hypothesis is rejected. Justify your answer.
  5. Explain whether or not the null hypothesis would be rejected if the test was carried out at the \(10 \%\) level of significance. \section*{Q uestion 1 continued} \section*{Q uestion 1 continued} \section*{Q uestion 1 continued}
Edexcel FS1 AS Specimen Q2
  1. The discrete random variable \(X\) has probability distribution given by
\(x\)- 10123
\(P ( X = x )\)\(c\)\(a\)\(a\)\(b\)\(c\)
The random variable \(Y = 2 - 5 X\)
Given that \(\mathrm { E } ( \mathrm { Y } ) = - 4\) and \(\mathrm { P } ( \mathrm { Y } \geqslant - 3 ) = 0.45\)
  1. find the probability distribution of X . Given also that \(\mathrm { E } \left( \mathrm { Y } ^ { 2 } \right) = 75\)
  2. find the exact value of \(\operatorname { Var } ( \mathrm { X } )\)
  3. Find \(\mathrm { P } ( \mathrm { Y } > \mathrm { X } )\) \section*{Q uestion 2 continued}
Edexcel FS1 AS Specimen Q3
  1. Two car hire companies hire cars independently of each other.
Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.
  1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
  2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
  3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
    In a random sample of 600 new cars produced at the factory,
    1. find the mean of the number of cars with a defect,
    2. find the variance of the number of cars with a defect.
    1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
    2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}
Edexcel FS1 AS Specimen Q4
  1. The discrete random variable \(X\) follows a Poisson distribution with mean 1.4
    1. Write down the value of
      1. \(\mathrm { P } ( \mathrm { X } = 1 )\)
      2. \(\mathrm { P } ( \mathrm { X } \leqslant 4 )\)
    The manager of a bank recorded the number of mortgages approved each week over a 40 week period.
    Number of mortgages approved0123456
    Frequency101674201
  2. Show that the mean number of mortgages approved over the 40 week period is 1.4 The bank manager believes that the Poisson distribution may be a good model for the number of mortgages approved each week. She uses a Poisson distribution with a mean of 1.4 to calculate expected frequencies as follows.
    Number of mortgages approved012345 or more
    Expected frequency9.86r9.674.511.58s
  3. Find the value of r and the value of s giving your answers to 2 decimal places. The bank manager will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution.
  4. Calculate the test statistic and state the conclusion for this test. State clearly the degrees of freedom and the hypotheses used in the test. \section*{Q uestion 4 continued} \section*{Q uestion 4 continued}
OCR FS1 AS 2021 June Q1
1 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.
  2. Find \(\mathrm { P } ( X = 11 )\).
  3. Find \(\mathrm { P } ( X \leqslant 8 )\).
  4. Find \(\operatorname { Var } ( X )\).
  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.
OCR FS1 AS 2021 June Q2
2 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 \(\operatorname { Var } ( 3 W + 6 )\). In this question you must show detailed reasoning.
    The random variable \(T\) has a binomial distribution. It is known that \(\mathrm { E } ( T ) = 5.625\) and the standard deviation of \(T\) is 1.875 . Find the values of the parameters of the distribution.
OCR FS1 AS 2021 June Q4
30 marks
4 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\). \end{table}
OCR FS1 AS 2017 December Q1
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.
    (a) State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
    (b) 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
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
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
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
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
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
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
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
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.
    (a) Write down the standard deviation of \(W\).
    (b) Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
  2. 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
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
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
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
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
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 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}