# Type I & Type II Errors(CIE A Level Maths: Probability & Statistics 2)

Author

Amber

Expertise

Maths

## 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)

#### Worked example

In the following scenarios, decide whether a Type I error or Type II error could have occurred

(i)
A farmer is testing for a change in crop growth after trying a new fertiliser. The test concludes that there is no evidence of change at the 5% significance level.

(ii)
A dentist’s receptionist believes that the waiting times have been reduced due to a new scheduling system. They conduct a hypothesis test and will reject the null hypothesis if no more than two customers wait more than ten minutes. Exactly two customers have to wait more than ten minutes.

#### Exam Tip

• Here are two tips if you cannot remember which error is which but are asked to calculate one on the exam:
• Look to see if you are given a new population parameter, this will be a Type II error.
• Check the number of marks, a Type I error is normally only 1 mark whilst a Type II error needs to be calculated and so will be more.

### Get unlimited access

to absolutely everything:

• Unlimited Revision Notes
• Topic Questions
• Past Papers