Find Type I error probability

A question is this type if and only if it asks to calculate or state the probability of a Type I error for a given hypothesis test, often involving finding P(reject H₀ | H₀ true).

2 questions · Standard +0.6

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CAIE S2 2023 March Q4
5 marks Standard +0.3
4 The number of accidents per 3-month period on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) has been 5.7. Following some changes to the road, the council carries out a hypothesis test to determine whether the value of \(\lambda\) has decreased. If there are fewer than 3 accidents in a randomly chosen 3 -month period, the council will conclude that the value of \(\lambda\) has decreased.
  1. Find the probability of a Type I error.
  2. Find the probability of a Type II error if the mean number of accidents per 3-month period is now actually 0.9 .
CAIE S2 2017 March Q4
7 marks Standard +0.8
4 At a doctors' surgery, the number of missed appointments per day has a Poisson distribution. In the past the mean number of missed appointments per day has been 0.9 . Following some publicity, the manager carries out a hypothesis test to determine whether this mean has decreased. If there are fewer than 3 missed appointments in a randomly chosen 5-day period, she will conclude that the mean has decreased.
  1. Find the probability of a Type I error.
  2. State what is meant by a Type I error in this context.
  3. Find the probability of a Type II error if the mean number of missed appointments per day is 0.2 .