Hypothesis Testing Lecture 4

Examples of various hypotheses The sodium content in Furresøen is x Sodium content in Furresøen is equal to the content in Madamsø The proportion of Turks in Aalborg is x % Proportion of Turks in Århus is the same as in Aalborg Average height of men in Sweden is the same as in Denmark

Basics Null hypothesis Alternative hypothesis Type I errors: Rejecting falsely Type II errors: Accepting falsely

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

Critical Region Assume We want to test if the sodium content 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 3.4 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 3.4 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 … Are 3.8 in the 95% CI ? Accept the hypothesis (mean = 3.8) on a 5% significance level That’s the same!!