Questions — Edexcel FS1 (49 questions)

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Edexcel FS1 2022 June Q5
  1. A random sample of 150 observations is taken from a geometric distribution with parameter 0.3
Estimate the probability that the mean of the sample is less than 3.45
Edexcel FS1 2022 June Q6
  1. The discrete random variable \(V\) has probability distribution
\(v\)234
\(\mathrm { P } ( V = v )\)\(\frac { 9 } { 25 }\)\(\frac { 12 } { 25 }\)\(\frac { 4 } { 25 }\)
  1. Show that the probability generating function of \(V\) is $$\mathrm { G } _ { V } ( t ) = t ^ { 2 } \left( \frac { 2 } { 5 } t + \frac { 3 } { 5 } \right) ^ { 2 }$$ The discrete random variable \(W\) has probability generating function $$\mathrm { G } _ { W } ( t ) = t \left( \frac { 2 } { 5 } t + \frac { 3 } { 5 } \right) ^ { 5 }$$
  2. Use calculus to find
    1. \(\mathrm { E } ( W )\)
    2. \(\operatorname { Var } ( W )\) Given that \(V\) and \(W\) are independent,
  3. find the probability generating function of \(X = V + W\) in its simplest form. The discrete random variable \(Y = 2 X + 3\)
  4. Find the probability generating function of \(Y\)
  5. Find \(\mathrm { P } ( Y = 15 )\)
Edexcel FS1 2022 June Q7
  1. A machine fills bags with flour. The weight of flour delivered by the machine into a bag, \(X\) grams, is normally distributed with mean \(\mu\) grams and standard deviation 30 grams. To check if there is any change to the mean weight of flour delivered by the machine into each bag, Olaf takes a random sample of 10 bags. The weight of flour, \(x\) grams, in each bag is recorded and \(\bar { x } = 1020\)
    1. Test, at the \(5 \%\) level of significance, \(\mathrm { H } _ { 0 } : \mu = 1000\) against \(\mathrm { H } _ { 1 } : \mu \neq 1000\)
    Olaf decides to alter the test so that the hypotheses are \(\mathrm { H } _ { 0 } : \mu = 1000\) and \(\mathrm { H } _ { 1 } : \mu > 1000\) but keeps the level of significance at 5\% He takes a second sample of size \(n\) and finds the critical region, \(\bar { X } > c\)
  2. Find an equation for \(c\) in terms of \(n\) When the true value of \(\mu\) is 1020 grams, the probability of making a Type II error is 0.0050 , to 2 significant figures.
  3. Calculate the value of \(n\) and the value of \(c\)
Edexcel FS1 2023 June Q1
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 2- 1013
\(\mathrm { P } ( X = x )\)0.25\(a\)\(b\)\(a\)0.30
where \(a\) and \(b\) are probabilities.
  1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 3.9\)
  2. find the value of \(a\) and the value of \(b\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
  3. Find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 3 \right)\)
Edexcel FS1 2023 June Q2
  1. Telephone calls arrive at a call centre randomly, at an average rate of 1.7 per minute. After the call centre was closed for a week, in a random sample of 10 minutes there were 25 calls to the call centre.
    1. Carry out a suitable test to determine whether or not there is evidence that the rate of calls arriving at the call centre has changed.
      Use a \(5 \%\) level of significance and state your hypotheses clearly.
    Only 1.2\% of the calls to the call centre last longer than 8 minutes.
    One day Tiang has 70 calls.
  2. Find the probability that out of these 70 calls Tiang has more than 2 calls lasting longer than 8 minutes. The call centre records show that \(95 \%\) of days have at least one call lasting longer than 30 minutes.
    On Wednesday 900 calls arrived at the call centre and none of them lasted longer than 30 minutes.
  3. Use a Poisson approximation to estimate the proportion of calls arriving at the call centre that last longer than 30 minutes.
Edexcel FS1 2023 June Q3
  1. In a class experiment, each day for 170 days, a child is chosen at random and spins a large cardboard coin 5 times and the number of heads is recorded.
    The results are summarised in the following table.
Number of heads012345
Frequency31045623812
Marcus believes that a \(\mathrm { B } ( 5,0.5 )\) distribution can be used to model these data and he calculates expected frequencies, to 2 decimal places, as follows
Number of heads012345
Expected frequency\(r\)26.56\(s\)\(s\)26.56\(r\)
  1. Find the value of \(r\) and the value of \(s\)
  2. Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the \(\mathrm { B } ( 5,0.5 )\) distribution is a good model for these data.
    You should state clearly your hypotheses, the test statistic and the critical value used. Nima believes that a better model for these data would be \(\mathrm { B } ( 5 , p )\)
  3. Find a suitable estimate for \(p\) To test her model, Nima uses this value of \(p\), to calculate expected frequencies as follows
    Number of heads012345
    Expected frequency2.0714.6541.4458.6341.4711.74
    The test statistic for Nima’s test is 1.62 (to 3 significant figures)
  4. State,
    1. giving your reasons, the degrees of freedom
    2. the critical value
      that Nima should use for a test at the 5\% significance level.
  5. With reference to Marcus' and Nima's test results, comment on
    1. the probability of the coin landing on heads,
    2. the independence of the spins of the coin. Give reasons for your answers.
Edexcel FS1 2023 June Q4
  1. There are 32 students in a class.
Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die. Estimate the probability that the mean number of rolls for the class is less than 27.2
Edexcel FS1 2023 June Q5
  1. A machine fills cartons with juice.
The amount of juice in a carton is normally distributed with mean \(\mu \mathrm { ml }\) and standard deviation 8 ml . A manager wants to test whether or not the amount of juice in the cartons, \(X \mathrm { ml }\), is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar { x } \mathrm { ml }\).
  1. Using a \(5 \%\) level of significance, find the critical region of \(\bar { X }\) for this test. State your hypotheses clearly. The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice \(\bar { y } \mathrm { ml }\) and uses a test with a critical region of $$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
  2. Find P (Type I error) for the Director's test. When \(\mu = 325 \mathrm { ml }\)
  3. find P (Type II error) for the test in part (a)
Edexcel FS1 2023 June Q6
  1. The discrete random variable \(X\) has probability generating function
$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$
  1. Specify the distribution of \(X\) A fair die is rolled repeatedly.
  2. Describe an outcome that could be modelled by the random variable \(X\)
  3. Use calculus and \(\mathrm { G } _ { X } ( t )\) to find
    1. \(\mathrm { E } ( X )\)
    2. \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
  4. Find the exact value of \(\mathrm { P } ( Y = 19 )\)
Edexcel FS1 2023 June Q7
  1. Each time a spinner is spun, the probability that it lands on red is 0.2
    1. Find the probability that the spinner lands on red
      1. for the 1st time on the 4th spin
      2. for the 3rd time on the 8th spin
      3. exactly 4 times during 10 spins
    Each time the spinner is spun, the probability that it lands on yellow is 0.4
    In a game with this spinner, a player must choose one of two events
    \(R\) is the event that the spinner lands on red for the \(\mathbf { 1 s t }\) time in at most 4 spins
    \(Y\) is the event that the spinner lands on yellow for the 3rd time in at most 7 spins
  2. Showing your calculations clearly, determine which of these events has the greater probability.
Edexcel FS1 2024 June Q1
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)- 10135
\(\mathrm { P } ( X = x )\)0.20.10.20.250.25
  1. Find \(\operatorname { Var } ( X )\)
  2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\)
Edexcel FS1 2024 June Q2
  1. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.
A 100-word piece of work completed by the secretary is selected at random.
  1. Find the probability that
    1. there are exactly 3 errors,
    2. there are fewer than 2 errors. After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.
  2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, find the critical region for a suitable test.
  3. Find P (Type I error) for the test in part (b)
Edexcel FS1 2024 June Q3
  1. Tisam took a survey of students' favourite colours. The results are summarised in the table below.
\multirow{2}{*}{}Colour
RedBlueGreenYellowBlackTotal
\multirow{3}{*}{Year group}1-534151422388
6-92332129884
10-12528198868
Total6275453919240
Tisam carries out a suitable test to see if there is any association between favourite colour and year group.
  1. Write down the hypotheses for a suitable test. For her table, Tisam only needs to check one cell to show that none of the expected frequencies are less than 5
    1. Identify this cell, giving your reason.
    2. Calculate the expected frequency for this cell. The test statistic for Tisam's test is 38.449
  3. Using a \(1 \%\) level of significance, complete the test. You should state your critical value and conclusion clearly.
Edexcel FS1 2024 June Q4
  1. Every morning Geethaka repeatedly rolls a fair, six-sided die until he rolls a 3 and then he stops. The random variable \(X\) represents the number of times he rolls the die each morning.
    1. Suggest a suitable model for the random variable \(X\)
    2. Show that \(\mathrm { P } ( X \leqslant 3 ) = \frac { 91 } { 216 }\)
    After 64 mornings Geethaka will calculate the mean number of times he rolled the die.
  2. Estimate the probability that the mean number of rolls is between 5.6 and 7.2 Nira wants to check Geethaka's die to decide whether or not the probability of rolling a 3 with his die is less than \(\frac { 1 } { 6 }\) Nira rolls the die repeatedly until she rolls a 3
    She obtains \(x = 16\)
  3. By carrying out a suitable test, determine what Nira's conclusion should be. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
Edexcel FS1 2024 June Q5
  1. Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
    Niluki monitors the production process and takes a random sample of \(n\) components.
    1. Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03
    Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\) In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
  2. Find the size of test \(\mathbf { A }\) In test B, Niluki uses \(n = 80\) and
    • if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
    • if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
    • if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
    • Find the size of test \(\mathbf { B }\)
    Given that the actual proportion of defective components is 0.06
    1. find the power of test \(\mathbf { A }\)
    2. find the expected number of components sampled using test \(\mathbf { B }\) Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
  4. suggest, giving your reasons, which test Niluki should use.
Edexcel FS1 2024 June Q6
  1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$
  1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
  2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
  3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
  4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)
Edexcel FS1 2024 June Q7
  1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)
When more than one game is played the games are independent.
Sam plays 20 games.
  1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
  2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
  3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
  4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
  5. Find, in terms of \(r\), Tessa's expected profit.
Edexcel FS1 Specimen Q1
  1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.
\section*{Q uestion 1 continued}
Edexcel FS1 Specimen Q2
  1. A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of a caller, chosen at random, being connected to the wrong agent is p.
The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent is 0.049 The call centre receives 1000 calls each day.
  1. Find the mean and variance of the number of wrongly connected calls a day.
  2. Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent.
  3. Explain why the approximation used in part (b) is valid. The probability that more than 6 calls each day are connected to the wrong agent using the binomial distribution is 0.8711 to 4 decimal places.
  4. Comment on the accuracy of your answer in part (b).
Edexcel FS1 Specimen Q3
  1. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins.
The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim B ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. He then calculates some of the expected frequencies, correct to 1 decimal place. \begin{table}[h]
Number of fake coins in each bag01234 or more
Observed frequency436226136
Expected frequency53.856.68.9
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]
    Number of fake coins in each bag01234 or more
    Observed frequency436226136
    Expected frequency44.555.733.212.54.1
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Explain why there are 2 degrees of freedom in this case.
  3. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel FS1 Specimen Q4
  1. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3
Estimate the probability that the mean of the sample is greater than 2.5
Edexcel FS1 Specimen Q5
  1. The probability of Richard winning a prize in a game at the fair is 0.15
Richard plays a number of games.
  1. Find the probability of Richard winning his second prize on his 8th game,
  2. State two assumptions that have to be made, for the model used in part (a) to be valid. M ary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable \(G\) represents the total number of games M ary plays.
  3. Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game.
Edexcel FS1 Specimen Q6
  1. The probability generating function of the discrete random variable \(X\) is given by
$$G _ { x } ( t ) = k \left( 3 + t + 2 t ^ { 2 } \right) ^ { 2 }$$
  1. Show that \(\mathrm { k } = \frac { 1 } { 36 }\)
  2. Find \(\mathrm { P } ( \mathrm { X } = 3 )\)
  3. Show that \(\operatorname { Var } ( \mathrm { X } ) = \frac { 29 } { 18 }\)
  4. Find the probability generating function of \(2 \mathrm { X } + 1\)
    \section*{Q uestion 6 continued} \section*{Q uestion 6 continued} \section*{Q uestion 6 continued}
Edexcel FS1 Specimen Q7
  1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.
Sam decides to spin the spinner 20 times and record the number of times it lands on red.
  1. Find the critical region for Sam's test.
  2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
  3. Find the critical region for Tessa's test.
  4. Find the size of Tessa's test.
    1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
    2. Find the power function for Tessa's test.
  6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)