IB DP Maths: AI HL复习笔记4.12.6 Type I & Type II Errors
Type I & Type II Errors
What are Type I & Type II errors?
- There are?four possible outcomes?of a hypothesis test:
- H0?is?false?and H0?is?rejected
- H0?is?true?and H0?is?not rejected
- The test is?accurate?for these two outcomes
- H0?is?true?and H0?is?rejected
- H0?is?false?and H0?is?not rejected
- The test has led to an?error?for these two outcomes
- A?Type I error?occurs when a hypothesis test gives?sufficient evidence to?reject H0?despite it?being true
- This is sometimes called a “false positive”
- In a court case this would be when the defendant is found?guilty despite being innocent
- A?Type II error?is when a hypothesis test gives?insufficient?evidence to reject H0?despite it?being false
- This is sometimes called a “false negative”
- In a court case this would be when the defendant is found?innocent despite being guilty
How do I find the probabilities of a Type I or Type II error occurring?
- You can calculate the probability of errors occurring?before a sample is taken
- The probabilities are?determined by the critical region
- Equally it is?determined by the?significance level?α%
- Critical regions are determined such that:
- They keep the?probability of a Type I error less than or equal?to?the?significance level
- They?maximise?the?probability of a Type I error
- The probability of a?Type I error?occurring is equal to the probability of?being in the critical region?if H0?were true
- P(Type I error) = P(being in the critical region | H0? is true)
- For a continuous distribution (normal,?t, χ2)
- P(Type I error) =?α%
- For a discrete distribution (binomial, Poisson)
- P(Type I error) ≤?α%
- The probability of a?Type II?error occurring is equal to the probability of?not being in the critical region?given the actual value of the population parameter
- P(Type II error) = P(not being in critical region | actual population parameter)
- Once a sample has been taken you can determine which error could have occurred
- If you?rejected H0?then you could have made a?Type I error
- If you?accepted H0?then you could have made a?Type II error
Can I reduce the probabilities of making a Type I or Type II error?
- You can?reduce?the probability of a?Type I?error by?reducing the significance level
- However this will?increase?the probability of a?Type II?error
- You can?reduce?the probability of a?Type II?error by?increasing the significance level
- However this will?increase?the probability of a?Type I error
- The only way to?reduce both?probabilities is by?increasing the size of the sample
Exam Tip
- If an exam question asks you to find the probability of a Type I or II error then double check that the test has not been carried out yet
- The examiner could test your understanding of errors by asking you to state which error could not have occurred once the test has been carried out
Worked Example
b)Given that Lucy actually hits the target 80% of the time with her left hand, find the probability of a Type II error.
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