Type I & Type II Errors (Edexcel A Level Further Maths): Revision Note

Type I & Type II Errors

What are Type I & Type II errors?

• There are four possible outcomes of a hypothesis test

  • Two good outcomes

    • H₀ was false and H₀ got rejected

    • H₀ was true and H₀ was not rejected

  • Two bad outcomes (errors)

    • H₀ was true but H₀ was rejected (a Type I error)

    • H₀ was false yet H₀ was not rejected (a Type II error)

• Type I errors occur 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

• Type II errors are 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 probability of a Type I or Type II error?

• The probability of a Type I error is the probability of being in the critical region (rejecting H₀) given H₀ was true

  • P(Type I error) = P(in the critical region | H₀ is true)

• The critical region itself was set up based on a significance level, α%, which assumed H₀ was true

  • So for continuous distributions (normal, )

    • P(Type I error) = α%

    • It's exactly equal to the significance level

    • You can often write this down with no calculation

  • For discrete distributions (binomial, Poisson, Geometric)

    • It's as close as you can get to α% whilst still being critical

    • So P(Type I error) is the actual significance level (≤ α%)

• The probability of a Type II error is the probability of not being in the critical region (not rejecting H₀) given H₀ was false 

  • You need to be given the actual population parameter to find this

    • For example, H₀ assumed but actually 

    • This is more helpful than just saying (though, in practice, harder to know)

  • P(Type II error) = P(not in the critical region | actual population parameter)

• Either error is not desirable in a hypothesis test

  • Ideally you want to know these probabilities before doing a test

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