Hypothesis test on Poisson rate

A question is this type if and only if it requires setting up and carrying out a one-tailed or two-tailed hypothesis test to determine whether observed count data provides evidence of a change in a Poisson rate.

4 questions · Standard +0.4

5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities
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CAIE S2 2024 March Q5
12 marks Standard +0.8
5 A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables \(G \sim \operatorname { Po } ( 0.10 )\) and \(B \sim \operatorname { Po } ( 0.15 )\) respectively.
  1. Find the probability that during a randomly chosen 2-day period no girls arrive late.
  2. Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3 .
  3. It is given that the values of \(\mathrm { P } ( G = r )\) and \(\mathrm { P } ( B = r )\) for \(r \geqslant 3\) are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys.
    Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.
  4. Test the teacher's claim at the \(5 \%\) significance level.
Edexcel S2 2017 June Q1
11 marks Standard +0.3
  1. At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.
    1. Find the probability that there are no signal failures on a randomly selected day.
    2. Find the probability that there is at least 1 signal failure on each of the next 3 days.
    3. Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
    Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded. A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
  2. State the hypotheses for this test.
  3. Find the largest value of \(f\) for which the null hypothesis should be rejected.
Edexcel S2 2024 June Q1
13 marks Standard +0.3
1 A garage sells tyres. The number of customers arriving at the garage to buy tyres in a 10-minute period is modelled by a Poisson distribution with mean 2
  1. Find the probability that
    1. fewer than 4 customers arrive to buy tyres in the next 10 minutes,
    2. more than 5 customers arrive to buy tyres in the next 10 minutes. The manager randomly selects 20 non-overlapping, 30-minute periods.
  2. Find the probability that there are between 4 and 7 (inclusive) customers arriving to buy tyres in exactly 15 of these 30-minute periods. The manager believes that placing an advert in the local paper will lead to a significant increase in the number of customers arriving at the garage.
    A week after the advert is placed, the manager randomly selects a 25 -minute period and finds that 10 customers arrive at the garage to buy tyres.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the manager's belief.
    State your hypotheses clearly.
  4. Explain why the Poisson distribution is unlikely to be valid for the number of tyres sold during a 10-minute period.
Edexcel S2 2001 June Q5
12 marks Standard +0.3
5. The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is
  1. exactly 4,
  2. more than 5 . Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
  3. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
  4. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased.