Questions — Edexcel FS1 AS (32 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}