Aims: To understand the difference between a one-tail and two tail test. To be able to formulate a null and alternative hypothesis. To be able to carry out a hypothesis test of a population mean from a sample greater than 30 with known variance. To know what Type I and Type II errors are. To be able to calculate Type I error for continuous data. Hypothesis Testing Lesson 2

In 1963 the average length of the populations middle finger was normally distributed with mean 81mm with standard deviation 3mm. By using yourselves and adding to the 15 measurements collected from another class, test at a 5% significance level to see if you think this is different nowadays! Measurements collected from a previous class in mm: Practical Example of Testing a Normal Mean Ho:H1:Ho:H1:

A question could also ask you to find the critical regions. What are they for this question?

Incorrect decision Correct decision Errors in hypothesis testing We can see that two of the outcomes result in an error. There are parallels in hypothesis testing. The defendant is found guilty when he is innocent The defendant is found innocent when he is guilty The defendant is found innocent when he is innocent The defendant is found guilty when he is guilty In English Law, a defendant is initially assumed innocent of the charges. The jury hears the evidence, and on the basis of this, makes a judgement about the defendant’s innocence or guilt. There are four possible outcomes to the trial:

Incorrect decision (type ΙI error) Correct decision Errors in hypothesis testing There are four possible outcomes to the hypothesis test: The null hypothesis is rejected when it is false The null hypothesis is rejected when it is true Incorrect decision (type Ι error) The null hypothesis is not rejected when it is false The null hypothesis is not rejected when it is true Correct decision

Summary A Type Ι error occurs if a true null hypothesis is rejected, i.e. P(Type Ι error) = P(reject H 0 | H 0 true) A Type ΙΙ error occurs if a false null hypothesis is accepted, i.e. P(Type ΙΙ error) = P(accept H 0 | H 0 false) In practice, the probability of a type I error is controlled by setting the s_______________ level of the test. Errors in hypothesis testing In this course you do need to be able to state and understand the two types of errors but you will not be asked to calculate the probability of a type ll error Typical question: State the Type l error for our previous experiment and explain in context of the question, what is meant by a ‘Type ll error’

The Swine Flu Errors Song H o : The patient does NOT have Swine Flu. H 1 : The patient does have Swine Flu. Reject, reject, reject H o (nought) When in fact it’s true This is called a type 1 error And not a thing to do! The second type we call Beta And accept H o But this time it is a fake And Swine flu they have caught. Some remember it as the positive error Some remember it as the negative error Your fine and well! Sorry you have Swine flu To tune: Row, row, row, your boat Wrong!

Errors in hypothesis testing Typical question: State the probability of making a Type l error for our practical experiment and explain in context of the question, what is meant by a ‘Type ll error’ Do exercise 5C questions 3 and 6 page 124

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