Questions FS1 AS (62 questions)

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Edexcel FS1 AS 2018 June Q1
  1. A researcher is investigating the distribution of orchids in a field. He believes that the Poisson distribution with a mean of 1.75 may be a good model for the number of orchids in each square metre. He randomly selects 150 non-overlapping areas, each of one square metre, and counts the number of orchids present in each square.
The results are recorded in the table below.
Number of orchids in
each square metre
0123456
Number of squares304235261160
He calculates the expected frequencies as follows
Number of orchids in
each square metre
012345More than 5
Number of squares26.0745.6239.9123.2810.193.57\(r\)
  1. Find the value of \(r\) giving your answer to 2 decimal places. The researcher will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution with mean 1.75
  2. State clearly the hypotheses required to test whether or not this Poisson distribution is a suitable model for these data. The test statistic for this test is 2.0 and the number of degrees of freedom to be used is 4
  3. Explain fully why there are 4 degrees of freedom.
  4. Stating your critical value clearly, determine whether or not these data support the researcher's belief. The researcher works in another field where the number of orchids in each square metre is known to have a Poisson distribution with mean 1.5 He randomly selects 200 non-overlapping areas, each of one square metre, in this second field, and counts the number of orchids present in each square.
  5. Using a Poisson approximation, show that the probability that he finds at least one square with exactly 6 orchids in it is 0.506 to 3 decimal places.
Edexcel FS1 AS 2018 June Q2
  1. The number of heaters, \(H\), bought during one day from Warmup supermarket can be modelled by a Poisson distribution with mean 0.7
    1. Calculate \(\mathrm { P } ( H \geqslant 2 )\)
    The number of heaters, \(G\), bought during one day from Pumraw supermarket can be modelled by a Poisson distribution with mean 3, where \(G\) and \(H\) are independent.
  2. Show that the probability that a total of fewer than 4 heaters are bought from these two supermarkets in a day is 0.494 to 3 decimal places.
  3. Calculate the probability that a total of fewer than 4 heaters are bought from these two supermarkets on at least 5 out of 6 randomly chosen days. December was particularly cold. Two days in December were selected at random and the total number of heaters bought from these two supermarkets was found to be 14
  4. Test whether or not the mean of the total number of heaters bought from these two supermarkets had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.
    VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel FS1 AS 2018 June Q3
  1. A fair six-sided black die has faces numbered \(1,2,2,3,3\) and 4
The random variable \(B\) represents the score when the black die is rolled.
  1. Write down the value of \(\mathrm { E } ( B )\) A white die has 6 faces numbered \(1,1,2,4,5\) and \(c\) where \(c > 5\)
    The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by
    \(w\)1245\(c\)
    \(\mathrm { P } ( W = w )\)\(a + b\)\(a\)0.3\(a\)\(b\)
    Greg and Nilaya play a game with these dice.
    Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
    The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
  2. Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\)
    Given that \(\mathrm { E } ( X ) = 2.6\)
  3. find the exact value of \(\operatorname { Var } ( X )\)
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FS1 AS 2018 June Q4
  1. Abram carried out a survey of two treatments for a plant fungus. The contingency table below shows the results of a survey of a random sample of 125 plants with the fungus.
\multirow{2}{*}{}Treatment
No actionPlant sprayed oncePlant sprayed every day
\multirow{3}{*}{Outcome}Plant died within a month151625
Plant survived for 1-6 months82510
Plant survived beyond 6 months7145
Abram calculates expected frequencies to carry out a suitable test. Seven of these are given in the partly-completed table below.
\multirow{2}{*}{}Treatment
No actionPlant sprayed oncePlant sprayed every day
\multirow{3}{*}{Outcome}Plant died within a month17.92
Plant survived for 1-6 months10.3218.9213.76
Plant survived beyond 6 months6.2411.448.32
The value of \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) for the 7 given values is 8.29
Test at the \(2.5 \%\) level of significance, whether or not there is an association between the treatment of the plants and their survival. State your hypotheses and conclusion clearly.
Edexcel FS1 AS 2019 June Q1
  1. A leisure club offers a choice of one of three activities to its 150 members on a Tuesday evening. The manager believes that there may be an association between the choice of activity and the age of the member and collected the following data.
\backslashbox{Age \(\boldsymbol { a }\) years}{Activity}BadmintonBowlsSnooker
\(a < 20\)933
\(20 \leqslant a < 40\)101014
\(40 \leqslant a < 50\)16155
\(50 \leqslant a < 60\)151311
\(a \geqslant 60\)4193
  1. Write down suitable hypotheses for a test of the manager's belief. The manager calculated expected frequencies to use in the test.
  2. Calculate the expected frequency of members aged 60 or over who choose snooker, used by the manager.
  3. Explain why there are 6 degrees of freedom used in this test. The test statistic used to test the manager's belief is 19.583
  4. Using a 5\% level of significance, complete the test of the manager's belief.
Edexcel FS1 AS 2019 June Q2
  1. A spinner used for a game is designed to give scores with the following probabilities
Score12346
Probability\(\frac { 3 } { 10 }\)\(\frac { 1 } { 10 }\)\(\frac { 1 } { 10 }\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)
The spinner is spun 80 times and the results are as follows
Score12346
Frequency15412418
Test, at the \(10 \%\) level of significance, whether or not the spinner is giving scores as it is designed to do. Show your working and state your hypotheses clearly.
Edexcel FS1 AS 2019 June Q3
  1. Andreia's secretary makes random errors in his work at an average rate of 1.7 errors every 100 words.
    1. Find the probability that the secretary makes fewer than 2 errors in the next 100 -word piece of work.
    Andreia asks the secretary to produce a 250 -word article for a magazine.
  2. Find the probability that there are exactly 5 errors in this article. Andreia offers the secretary a choice of one of two bonus schemes, based on a random sample of 40 pieces of work each consisting of 100 words. In scheme \(\mathbf { A }\) the secretary will receive the bonus if more than 10 of the 40 pieces of work contain no errors. In scheme \(\mathbf { B }\) the bonus is awarded if the total number of errors in all 40 pieces of work is fewer than 56
  3. Showing your calculations clearly, explain which bonus scheme you would advise the secretary to choose. Following the bonus scheme, Andreia randomly selects a single 500 -word piece of work from the secretary to test if there is any evidence that the secretary's rate of errors has decreased.
  4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test.
Edexcel FS1 AS 2019 June Q4
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 3- 1124
\(\mathrm { P } ( X = x )\)\(q\)\(\frac { 7 } { 30 }\)\(\frac { 7 } { 30 }\)\(q\)\(r\)
where \(q\) and \(r\) are probabilities.
  1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
  2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
  3. find the value of \(q\) and the value of \(r\)
  4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)
Edexcel FS1 AS 2020 June Q1
  1. A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.
    1. Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day.
    The company has enough staff to respond to 28 call-outs in an 8 -hour working day.
  2. Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places. In a random sample of 100 working days each of 8 hours,
    1. find the expected number of days that the company receives more than 28 call-outs,
    2. find the standard deviation of the number of days that the company receives more than 28 call-outs,
    3. use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.
Edexcel FS1 AS 2020 June Q2
  1. In an experiment, James flips a coin 3 times and records the number of heads. He carries out the experiment 100 times with his left hand and 100 times with his right hand.
\multirow{2}{*}{}Number of heads
0123
Left hand7294222
Right hand13353616
  1. Test, at the \(5 \%\) level of significance, whether or not there is an association between the hand he flips the coin with and the number of heads. You should state your hypotheses, the degrees of freedom and the critical value used for this test.
  2. Assuming the coin is unbiased, write down the distribution of the number of heads in 3 flips.
  3. Carry out a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether or not the distribution you wrote down in part (b) is a suitable model for the number of heads obtained in the 200 trials of James' experiment. You should state your hypotheses, the degrees of freedom and the critical value used for this test.
Edexcel FS1 AS 2020 June Q3
  1. The probability distribution of the discrete random variable \(X\) is
$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3
\frac { m } { 2 x } & \text { for } x = 6 \text { and } 9
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)
Edexcel FS1 AS 2020 June Q4
  1. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8
The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3
  1. Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. Given that exactly 12 cyclists pass the point in a 10 -minute interval,
  2. find the probability that at least 11 are travelling eastbound. After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14
  3. Test, at the \(5 \%\) level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
    State your hypotheses clearly.
Edexcel FS1 AS 2021 June Q1
  1. Flobee sells tomato seeds in packets, each containing 40 seeds. Flobee advertises that only 4\% of its tomato seeds do not germinate.
Amodita is investigating the germination of Flobee's tomato seeds. She plants 125 packets of Flobee's tomato seeds and records the number of seeds that do not germinate in each packet.
Number of seeds that do not germinate0123456 or more
Frequency153538221050
Amodita wants to test whether the binomial distribution \(\mathrm { B } ( 40,0.04 )\) is a suitable model for these data. The table below shows the expected frequencies, to 2 decimal places, using this model.
Number of seeds that do not germinate012345 or more
Expected Frequency24.4240.70\(r\)17.456.73\(s\)
  1. Calculate the value of \(r\) and the value of \(s\)
  2. Stating your hypotheses clearly, carry out the test at the \(5 \%\) level of significance. You should state the number of degrees of freedom, critical value and conclusion clearly. Amodita believes that Flobee should use a more realistic value for the percentage of their tomato seeds that do not germinate.
    She decides to test the data using a new model \(\mathrm { B } ( 40 , p )\)
  3. Showing your working, suggest a more realistic value for \(p\)
Edexcel FS1 AS 2021 June Q2
  1. Rowan and Alex are both check-in assistants for the same airline. The number of passengers, \(R\), checked in by Rowan during a 30-minute period can be modelled by a Poisson distribution with mean 28
    1. Calculate \(\mathrm { P } ( R \geqslant 23 )\)
    The number of passengers, \(A\), checked in by Alex during a 30-minute period can be modelled by a Poisson distribution with mean 16, where \(R\) and \(A\) are independent. A randomly selected 30-minute period is chosen.
  2. Calculate the probability that exactly 42 passengers in total are checked in by Rowan and Alex. The company manager is investigating the rate at which passengers are checked in. He randomly selects 150 non-overlapping 60-minute periods and records the total number of passengers checked in by Rowan and Alex, in each of these 60-minute periods.
  3. Using a Poisson approximation, find the probability that for at least 25 of these 60-minute periods Rowan and Alex check in a total of fewer than 80 passengers. On a particular day, Alex complains to the manager that the check-in system is working slower than normal. To see if the complaint is valid the manager takes a random 90-minute period and finds that the total number of people Rowan checks in is 67
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the system is working slower than normal. You should state your hypotheses and conclusion clearly and show your working.
Edexcel FS1 AS 2021 June Q3
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 3- 2- 1025
\(\mathrm { P } ( X = x )\)0.30.150.10.150.10.2
  1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 8.79\)
  2. find \(\mathrm { E } \left( X ^ { 2 } \right)\) The discrete random variable \(Y\) has probability distribution
    \(y\)- 2- 1012
    \(\mathrm { P } ( Y = y )\)\(3 a\)\(a\)\(b\)\(a\)\(c\)
    where \(a\), \(b\) and \(c\) are constants.
    For the random variable \(Y\) $$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
  3. Find the value of \(a\), the value of \(b\) and the value of \(c\) The random variable \(W = Y - X\) where \(Y\) and \(X\) are independent.
    The random variable \(T = 3 W - 8\)
  4. Calculate \(\mathrm { P } ( W > T )\)
Edexcel FS1 AS 2021 June Q4
  1. Charlie carried out a survey on the main type of investment people have.
The contingency table below shows the results of a survey of a random sample of people.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Main type of investment
\cline { 3 - 5 } \multicolumn{2}{c|}{}BondsCashStocks
\multirow{2}{*}{Age}\(25 - 44\)\(a\)\(b - e\)\(e\)
\cline { 2 - 5 }\(45 - 75\)\(c\)\(d - 59\)59
  1. Find an expression, in terms of \(a , b , c\) and \(d\), for the difference between the observed and the expected value \(( O - E )\) for the group whose main type of investment is Bonds and are aged 45-75
    Express your answer as a single fraction in its simplest form. Given that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62\) for this information,
  2. test, at the \(5 \%\) level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.
Edexcel FS1 AS 2022 June Q1
  1. Stuart is investigating a treatment for a disease that affects fruit trees. He has 400 fruit trees and applies the treatment to a random sample of these trees. The remainder of the trees have no treatment. He records the number of years, \(y\), that each fruit tree remains free from this disease.
The results are summarised in the table below.
\cline { 3 - 3 } \multicolumn{2}{c|}{}Treatment
\cline { 3 - 4 } \multicolumn{2}{c|}{}AppliedNot applied
\multirow{3}{*}{
Number of years free
from this disease
}		\(y < 1\)1525
\cline { 2 - 4 }\(1 \leqslant y < 2\)3561
\cline { 2 - 4 }\(2 \leqslant y\)124140
The data are to be used to determine whether or not there is an association between the application of the treatment and the number of years that a fruit tree remains free from this disease.
  1. Calculate the expected frequencies for
    1. Applied and \(y < 1\)
    2. Not applied and \(1 \leqslant y < 2\) The value of \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) for the other four classes is 2.642 to 3 decimal places.
  2. Test, at the \(5 \%\) level of significance, whether or not there is an association between the application of the treatment and the number of years a fruit tree remains free from this disease. You should state your hypotheses, test statistic, critical value and conclusion clearly.
Edexcel FS1 AS 2022 June Q2
  1. Xena catches fish at random, at a constant rate of 0.6 per hour.
    1. Find the probability that Xena catches exactly 4 fish in a 5 -hour period.
    The probability of Xena catching no fish in a period of \(t\) hours is less than 0.16
  2. Find the minimum value of \(t\), giving your answer to one decimal place. Independently of Xena, Zion catches fish at random with a mean rate of 0.8 per hour.
    Xena and Zion try using new bait to catch fish. The number of fish caught in total by Xena and Zion after using the new bait, in a randomly selected 4-hour period, is 12
  3. Use a suitable test to determine, at the \(5 \%\) level of significance, whether or not there is evidence that the rate at which fish are caught has increased after using the new bait. State your hypotheses clearly and the \(p\)-value used in your test.
Edexcel FS1 AS 2022 June Q3
  1. In a game, a coin is spun 5 times and the number of heads obtained is recorded. Tao suggests playing the game 20 times and carrying out a chi-squared test to investigate whether the coin might be biased.
    1. Explain why playing the game only 20 times may cause problems when carrying out the test.
    Chris decides to play the game 500 times. The results are as follows
    Number of heads012345
    Observed frequency2279318114651
    Chris decides to test whether or not the data can be modelled by a binomial distribution, with the probability of a head on each spin being 0.6 She calculates the expected frequencies, to 2 decimal places, as follows
    Number of heads012345
    Expected frequency5.1238.40115.20172.80129.6038.88
  2. State the number of degrees of freedom in Chris' test, giving a reason for your answer.
  3. Carry out the test at the \(5 \%\) level of significance. You should state your hypotheses, test statistic, critical value and conclusion clearly.
  4. Showing your working, find an alternative model which would better fit Chris’ data.
Edexcel FS1 AS 2022 June Q4
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)0236
\(\mathrm { P } ( X = x )\)\(p\)0.25\(q\)0.4
  1. Find in terms of \(q\)
    1. \(\mathrm { E } ( X )\)
    2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
  2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\)
    Given that games are independent,
  3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
  4. Find the probability that Bobby wins at least \(\pounds 10\)
Edexcel FS1 AS 2023 June Q1
  1. The discrete random variable \(X\) has the following distribution
\(x\)01234
\(\mathrm { P } ( X = x )\)\(r\)\(k\)\(\frac { k } { 2 }\)\(\frac { k } { 3 }\)\(\frac { k } { 4 }\)
where \(r\) and \(k\) are positive constants.
The standard deviation of \(X\) equals the mean of \(X\)
Find the exact value of \(r\)
Edexcel FS1 AS 2023 June Q2
  1. A bag contains a large number of balls, all of the same size and weight. The balls are coloured Red, Blue or Yellow.
Jasmine asks each child in a group of 150 children to close their eyes, select a ball from the bag and show it to her. The child then replaces the ball and repeats the process a second time. If both balls are the same colour the child receives a prize.
The results are given in the table below.
\backslashbox{2nd colour}{1st colour}RedBlueYellowTotal
Red31111860
Blue810927
Yellow2193363
Total603060150
Jasmine carries out a test, at the \(5 \%\) level of significance, to see whether or not the colour of the 2nd ball is independent of the colour of the 1st ball.
  1. Calculate the expected frequencies for the cases where both balls are the same colour. The test statistic Jasmine obtained was 12.712 to three decimal places.
  2. Use this value to complete the test, stating the critical value and conclusion clearly. With reference to your calculations in part (a) and the nature of the experiment, (c) give a plausible reason why Jasmine may have obtained her conclusion in part (b).
Edexcel FS1 AS 2023 June Q3
  1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.
The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907
  1. Find the value of \(A\) The tablecloths are sold in packets of 20
    A randomly selected packet is taken.
  2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
    The random variable \(X\) represents the number of these tablecloths that have no faults.
  3. Find
    1. \(\mathrm { E } ( X )\)
    2. \(\operatorname { Var } ( X )\)
  4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
    The piece of cloth has area 30 square metres and is found to have 6 faults.
  5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
  6. Write down the \(p\)-value for the test used in part (e).
Edexcel FS1 AS 2023 June Q4
  1. Table 1 below shows the number of car breakdowns in the Snoreap district in each of 60 months.
\begin{table}[h]
Number of car
breakdowns
012345
Frequency1211191431
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Anja believes that the number of car breakdowns per month in Snoreap can be modelled by a Poisson distribution. Table 2 below shows the results of some of her calculations. \begin{table}[h]
Number of car breakdowns01234\(\geqslant 5\)
Observed frequency (O)1211191431
Expected frequency ( \(\mathbf { E } _ { \mathbf { i } }\) )9.929.644.34
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. State suitable hypotheses for a test to investigate Anja's belief.
  2. Explain why Anja has changed the label of the final column to \(\geqslant 5\)
  3. Showing your working clearly, complete Table 2
  4. Find the value of \(\frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) when the number of car breakdowns is
    1. 1
    2. 3
  5. Explain why Anja used 3 degrees of freedom for her test. The test statistic for Anja's test is 6.54 to 2 decimal places.
  6. Stating the critical value and using a \(5 \%\) level of significance, complete Anja's test.
Edexcel FS1 AS 2024 June Q1
  1. Sharma believes that each computer game he sells appeals equally to all age ranges.
To investigate this, he takes a random sample of 100 people who play these games and asks them which of the games \(A , B\) or \(C\) they prefer.
The results are summarised in the table below.
Computer game\(A\)\(B\)\(C\)
\multirow{3}{*}{Age range}\(< 20\)8156
\cline { 2 - 5 }\(20 - 30\)21129
\cline { 2 - 5 }\(> 30\)61013
  1. Write down hypotheses for a suitable test to assess Sharma's belief.
  2. For the test, calculate the expected frequency for
    1. those players aged under 20 who prefer game \(C\)
    2. those players aged between 20 and 30 who prefer game \(A\)
  3. State the degrees of freedom of the test statistic for this test. Sharma correctly calculates the test statistic for this test to be 11.542 (to 3 decimal places).
  4. Using a \(5 \%\) significance level, and stating your critical value, comment on Sharma's belief.