Hypothesis Testing Lecture 3

Examples of various hypotheses Average salary in Copenhagen is larger than in Bælum Sodium content in Furresøen is equal to the content in Madamsø Proportion of Turks in Århus is the same as in Aalborg Average height of men in Sweden is the same as in Denmark The average temperature is increasing over time

Formulation of hypothesis Assume we are interested in a parameter Θ (e.g. the mean of the data). Let Θ 0 be a number. There are three different kinds of hypotheses: H 0 : Θ = Θ 0 H 0 : Θ ≥ Θ 0 H 0 : Θ ≤ Θ H A : Θ ≠ Θ 0 H A : Θ Θ 0 H 0 is called the null hypothesis. H A is called the alternative hypothesis.

Examples of various hypotheses Average salary in Copenhagen is larger than in Bælum H 0 : μ C ≥ μ B. H A : μ C < μ B. Sodium content in Furresøen is equal to the content in Madamsø H 0 : μ F = μ M. H A : μ F ≠ μ M. Proportion of Turks in Århus is the same as in Aalborg H 0 : P Å = P A. H A : P Å ≠ P A. Average height of men in Sweden is the same as in Denmark H 0 : μ S = μ D. H A : μ S ≠ μ D. The average temperature is increasing over time H 0 : μ time 1 ≥ μ time 2. H A : μ time 1 < μ time 2 if time 1 ≥ time 2.

COMPARE SMALL DIFFERENCE BIG DIFFERENCEE NOT EQUAL MEANS EQUAL MEANS NORMAL DISTRIBUTION (average height in Sweden and Denmark)

BINOMIAL DISTRIBUTION (Proportion of Turks in Århus and Aalborg) BIG OR NOT?

The Test Procedure Formulate a HYPOTHESIS!

Numerically bigger than Does the data support the hypothesis or not?

Types of errors Type I error: Rejecting falsely. Type II error: Accepting falsely. DecisionH 0 is trueH 0 is false Reject H 0 Type I errorNo error Accept H 0 No errorType II error Ideally we would like a test where it is difficult to make errors.

Unfortunately If you make a test where it is difficult to make a Type I error it is easy to make a Type II error and the other way around

Level of significance So we want to construct a way to decide to ACCEPT or REJECT the hypothesis based on data in a way such that

This sounds really technical!!! Hmm I don’t like this at all!

Critical Region Assume We want to test if the sodium contest here is approx 3.8 units We have data y 1, …, y n We have calculated average and SE. Support that content is 3.8 Support that content is < 3.8 Support that content is > 3.8

What do we know? If the content is 3.8 then the average is normally distributed with mean 3.8 With probability of 95% is the average less than 2*SE from 3.8 If the true content is 3.8 then the average is in the red area with prob 5%

Test: The hypothesis is that the true content is 3.8 Estimate mean and SE. The critical region is If the average is in the critical area then reject the hypothesis else accept Significance level Prob(Type I error) = 5 %

Alternative approach Can we give a number telling us to what extend the observations support the hypothesis? Yes, of course! Why do you think I asked? Hmmm Supports hypothesis Here we should definitely reject

If the true content is 3.8 then and Assume that we observe an average of 3.8 and SE = 0.1 Then what?

What is the probability of observing this??? 95% of data sets will have an average in this area (mean +/- 2 SE) Assume we obtain an average of 3.8 and standard error SE = 0.1 and the true concentration is 3.8

P-value

Summing Up A Statistical test can be 1.On a 5% significance level 2.By calculating the p-value

Hypothesis about the Mean 1.Is the concentration 3.8? 2.Is the proprotion of Turks in Århus 7.5% Normal Distribution Binomial Distribution

Sodium 1.Are data normal? 2.Estimate average and standard error 3.Calculate 4.Is t bigger than 2 (numerically)? OR 5.Calculate p-value

Turks 1.Are data binomial? 2.Calculate proportion p and standard error 3.Calculate 4.Is t bigger than 2 (numerically)?

Last slide before the end Are 3.8 in the 95% CI ? Accept the hypothesis (mean = 3.8) on a 5% significance level That’s the same!!

The End