Questions — Edexcel (9670 questions)

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Edexcel FS1 2021 June Q1
7 marks Moderate -0.3
  1. Kelly throws a tetrahedral die \(n\) times and records the number on which it lands for each throw.
She calculates the expected frequency for each number to be 43 if the die was unbiased.
The table below shows three of the frequencies Kelly records but the fourth one is missing.
Number1234
Frequency473436\(x\)
  1. Show that \(x = 55\) Kelly wishes to test, at the \(5 \%\) level of significance, whether or not there is evidence that the tetrahedral die is unbiased.
  2. Explain why there are 3 degrees of freedom for this test.
  3. Stating your hypotheses clearly and the critical value used, carry out the test.
Edexcel FS1 2021 June Q2
14 marks Challenging +1.2
  1. On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.
    1. Show that the probability that on a weekday at least 2 calls are received by the garage in a 30 -minute period is 0.888 to 3 decimal places.
    2. Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday.
    The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.
    She records the number of calls received in each of these 30-minute periods.
  2. Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30 -minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.
  3. Explain why the Poisson approximation may be reasonable in this case. The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.
  4. Write down the hypotheses for this test. The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.
  5. Carry out the test using a \(5 \%\) level of significance.
Edexcel FS1 2021 June Q3
4 marks Standard +0.8
  1. A courier delivers parcels. The random variable \(X\) represents the number of parcels delivered successfully each day by the courier where \(X \sim \mathrm {~B} ( 400,0.64 )\)
A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { 100 }\) is taken.
Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257
Edexcel FS1 2021 June Q4
10 marks
  1. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
    Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below.
\(n\)012345
\(\mathrm { P } ( N = n )\)\(a\)0.20.050.25\(b\)\(c\)
Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)
  1. show that \(\operatorname { Var } ( N ) = 2.76\) The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £1 Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
  2. calculate the expected entry fee per member. Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.
  3. State a limitation of the Poisson distribution in this case.
Edexcel FS1 2021 June Q5
18 marks Standard +0.8
  1. Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.
    Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.
The probability of Asha getting a red counter on any one draw is 0.07
  1. Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.
  2. Find the probability that Asha gets a red counter for the second time on her 9th draw. The probability of Davinda getting a red counter on any one draw is \(p\). Davinda draws counters until she gets \(n\) red counters. The random variable \(D\) is the number of counters Davinda draws. Given that the mean and the standard deviation of \(D\) are 4400 and 660 respectively,
  3. find the value of \(p\). Jerry believes that his bag contains a smaller proportion of red counters than Asha's bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable \(J\) to be the number of counters drawn up to and including the first red counter.
  4. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. Jerry gets a red counter for the first time on his 34th draw.
  5. Giving a reason for your answer, state whether or not there is evidence that Jerry's bag contains a smaller proportion of red counters than Asha’s bag. Given that the probability of Jerry getting a red counter on any one draw is 0.011
  6. show that the power of the test is 0.702 to 3 significant figures.
Edexcel FS1 2021 June Q6
14 marks Standard +0.8
  1. The probability generating function of the random variable \(X\) is
$$\mathrm { G } _ { X } ( t ) = k ( 1 + 2 t ) ^ { 5 }$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 243 }\)
  2. Find \(\mathrm { P } ( X = 2 )\)
  3. Find the probability generating function of \(W = 2 X + 3\) The probability generating function of the random variable \(Y\) is $$\mathrm { G } _ { Y } ( t ) = \frac { t ( 1 + 2 t ) ^ { 2 } } { 9 }$$ Given that \(X\) and \(Y\) are independent,
  4. find the probability generating function of \(U = X + Y\) in its simplest form.
  5. Use calculus to find the value of \(\operatorname { Var } ( U )\)
Edexcel FS1 2021 June Q7
8 marks Challenging +1.8
  1. A manufacturer has a machine that produces lollipop sticks.
The length of a lollipop stick produced by the machine is normally distributed with unknown mean \(\mu\) and standard deviation 0.2 Farhan believes that the machine is not working properly and the mean length of the lollipop sticks has decreased.
He takes a random sample of size \(n\) to test, at the 1\% level of significance, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 15 \quad \mathrm { H } _ { 1 } : \mu < 15$$
  1. Write down the size of this test. Given that the actual value of \(\mu\) is 14.9
    1. calculate the minimum value of \(n\) such that the probability of a Type II error is less than 0.05
      Show your working clearly.
    2. Farhan uses the same sample size, \(n\), but now carries out the test at a \(5 \%\) level of significance. Without doing any further calculations, state how this would affect the probability of a Type II error.
Edexcel FS1 2022 June Q1
9 marks Standard +0.3
  1. A researcher is investigating the number of female cubs present in litters of size 4 He believes that the number of female cubs in a litter can be modelled by \(\mathrm { B } ( 4,0.5 )\) He randomly selects 100 litters each of size 4 and records the number of female cubs. The results are recorded in the table below.
Number of female cubs01234
Observed number of litters103333159
He calculated the expected frequencies as follows
Number of female cubs01234
Expected number of litters6.25\(r\)\(s\)\(r\)6.25
  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 number of female cubs in a litter can be modelled by \(\mathrm { B } ( 4,0.5 )\) You should clearly state your hypotheses and the critical value used.
Edexcel FS1 2022 June Q2
9 marks Standard +0.3
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 5- 105\(b\)
\(\mathrm { P } ( X = x )\)0.30.250.10.150.2
where \(b\) is a constant and \(b > 5\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(b\) Given that \(\operatorname { Var } ( X ) = 34.26\)
  2. find the value of \(b\)
  3. Find \(\mathrm { P } \left( X ^ { 2 } < 2 - 3 X \right)\)
Edexcel FS1 2022 June Q3
14 marks Standard +0.8
  1. During the summer, mountain rescue team \(A\) receives calls for help randomly with a rate of 0.4 per day.
    1. Find the probability that during the summer, mountain rescue team \(A\) receives at least 19 calls for help in 28 randomly selected days.
    The leader of mountain rescue team \(A\) randomly selects 250 summer days from the last few years.
    She records the number of calls for help received on each of these days.
  2. Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team \(A\). Mountain rescue team \(A\) believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8
  3. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly. During the summer, mountain rescue team \(B\) receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team \(A\). The random variable \(C\) is the total number of calls for help received by mountain rescue teams \(A\) and \(B\) during a period of \(n\) days in the summer.
    On a Monday in the summer, mountain rescue teams \(A\) and \(B\) each receive a call for help. Given that over the next \(n\) days \(\mathrm { P } ( C = 0 ) < 0.001\)
  4. calculate the minimum value of \(n\)
  5. Write down an assumption that needs to be made for the model to be appropriate.
Edexcel FS1 2022 June Q4
13 marks Standard +0.3
  1. In a game a spinner is spun repeatedly. When the spinner is spun, the probability of it landing on blue is 0.11
    1. Find the probability that the spinner lands on blue
      1. for the first time on the 6th spin,
      2. for the first time before the 6th spin,
      3. exactly 4 times during the first 6 spins,
      4. for the 4th time on or before the 6th spin.
    Zac and Izana play the game. They take turns to spin the spinner. The winner is the first one to have the spinner land on blue. Izana spins the spinner first.
  2. Show that the probability of Zac winning is 0.471 to 3 significant figures.
Edexcel FS1 2022 June Q5
5 marks Standard +0.8
  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
14 marks Standard +0.3
  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
11 marks Challenging +1.2
  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
9 marks Standard +0.3
  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
11 marks Standard +0.8
  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
15 marks Standard +0.8
  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
6 marks Challenging +1.2
  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
8 marks Standard +0.3
  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
13 marks Challenging +1.2
  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
13 marks Standard +0.3
  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
6 marks Standard +0.3
  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
7 marks Standard +0.8
  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
6 marks Standard +0.3
  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
12 marks Standard +0.3
  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.