Normal approximation to summed Poisson

A question is this type if and only if it requires using a Normal distribution to approximate a Poisson (or sum of Poissons) over a large time/space period, typically involving a continuity correction.

3 questions · Standard +0.8

5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02n Sum of Poisson variables: is Poisson
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CAIE S2 2024 November Q5
11 marks Challenging +1.2
5 A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions \(\operatorname { Po } ( 3.7 )\) and \(\operatorname { Po } ( 2.6 )\) respectively. A bag of sweets is chosen at random.
  1. Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5 .
  2. Find the probability that the total number of lemon and orange sweets in the bag is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-06_2725_47_107_2002} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-07_2716_29_107_22} 10 bags of sweets are chosen at random.
  3. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.
CAIE S2 2006 November Q4
7 marks Standard +0.3
4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.
  1. Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.
  2. Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.
Edexcel FS1 AS 2019 June Q3
13 marks Standard +0.8
  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.