CIE A Level Maths: Probability & Statistics 2复习笔记3.1.2 Type I & Type II Errors

Any hypothesis test will only provide evidence about whether a parameter has changed or not. A conclusion can not claim with certainty whether to accept or reject the null hypothesis as the test is based on probability, and therefore errors are possible.

What is a Type I error?

• A Type I error occurs when the null hypothesis is rejected incorrectly
  • In order for a Type I error to happen, the null hypothesis must have been rejected
• If a Type I error has been made, the hypothesis test has provided evidence that there is a change when in fact there is not a change
  • Think about the impact of this in some scenarios
    • For example a test saying that a student had cheated in an exam when in fact they had not
• The probability of a Type I error occurring in any hypothesis test is the same as the probability of rejecting a true null hypothesis
  • This is the probability of the observed value being at least as extreme as the critical value(s)
  • It is the same or a little bit less than the significance level
• In a true hypothesis test you would not need to calculate the probability of a Type I error as it would be the same as the actual significance level

What is a Type II error?

• A Type II error occurs when the null hypothesis is accepted incorrectly
  • In order for a Type II error to happen, the null hypothesis must not have been rejected
• If a Type II error has been made, the hypothesis test has provided evidence that there is no change when in fact there was a change
  • Think about the impact of this in some scenarios
    • For example a test saying that a car’s brakes have not worn down, when in fact they have
• To find probability of a Type II error occurring in any hypothesis test you would need to be given the true value of the population parameter being tested
  • For example, you would be given the true probability of the event occurring or the true population mean
  • The probability of a Type II error would be the probability of the observed value being outside of the rejection region, given the true value of the population parameter

Can the probabilities of making the errors be manipulated?

• It is possible to reduce the probability of making a Type I error by reducing the significance level before carrying out the test
  • However, this would decrease the size of the rejection region and therefore could increase the probability of a Type II error
• It is possible to reduce the probability of making a Type II error by increasing the significance level before carrying out the test
  • This would increase the size of the rejection region, making it easier to reject the null hypothesis
  • As the probability of rejecting the null hypothesis has increased, this would increase the probability of making a Type I error
• Before setting the significance level a researcher could consider which error they would want to reduce the likelihood of
  • For example, if the test is for a company advertising that their product works 90% of the time, but customers believe it may be less than this:
    • the company would want to reduce the probability of a Type I error (incorrectly declaring a change)
    • the customers would want to reduce the probability of a Type II error (incorrectly declaring no change)