1. Elementary statistics for foresters Lecture 5 Socrates/Erasmus Program @ WAU Spring semester 2005/2006

2. Statistical tests

3. Why using tests? Statistical hypotheses Errors in tests Test of significance Examples of tests

4. Why do we use tests? We work with samples We want to know about populations Sample = uncertainty So: we need a tool to be able to answer questions about population based on results from the sample Some examples...

5. Statistical hypotheses Hypothesis: it is a statement about parameters or variable distribution of population Hypothesis refers to a parameter – parametric hypothesis Hypothesis refers to a distribution – non- parametric hypothesis

6. Parametric hypotheses They are usually written as a short equation, e.g. μ = 44 μ 1 = μ 2 σ 1 = σ 2

7. Non-parametric hypotheses Usually written as a sentence, such as e.g. –„the distribution of the x variable in the population follows the normal distribution” –„samples were drawn from populations having the same distributions” –... Used not only exactly for distributions

8. Statistical hypotheses Null hypothesis – a hypothesis being tested during the testing procedure Alternative hypothesis – a reserve hypothesis used when the null hypothesis is not true –These hypotheses can be both: parametric and non-parametric.

9. Statistical hypotheses H 0 : μ = 44 H 0 : μ 1 = μ 2 H 0 : the distribution of the „x" variable follows the normal distribution

10. Statistical hypotheses H 1 : μ ≠ 44 H 1 : μ 1 ≠ μ 2 H 1 : the distribution of the „x" variable doesn’t follow the normal distribution

11. Errors in tests The hypothesis can be: true or false The result of the test can be: accept or reject the null hypothesis All possible cases are: –H 0 is true, test accepts the hypothesis –H 0 is true, test rejects the hypothesis –H 0 is false, test accepts the hypothesis –H 0 is false, test rejects the hypothesis

12. Errors in test In two cases we have a bad scenario: –H 0 is true, test rejects the hypothesis –H 0 is false, test accepts the hypothesis In these cases we have an error in using a statistical test All cases can be shown in the table:

13. Errors in tests Hypothesis / decisionAcceptReject true OK Type I error / error of the 1 st kind falseType II error / error of the 2 nd kind OK

14. Errors in tests Hypothesis / decisionAcceptReject true OK alpha error falsebeta error OK

15. How to avoid errors? test construction: use only tests rejecting hypotheses or saying that this is not enough to reject it. By doing so you can avoid type II errors, choose small significance level. (Test of significance)

16. Test of significance scheme formulate H 0 and H 1, sample the population(s), calculate a statistics for a given test (such statistics is also a variable having it's distribution if the null hypothesis is true), compare the calculated statistics with a critical value of the statistics for a given significance level reject the null hypothesis is rejected or state, that "we can't reject the null hypothesis for a given significance level α”

17. Test of significance in practice When using any statistical software – the end of the test is different. Instead of comparison of calculated test statistics with its theoretical value for a given significance level – p-value („critical significance level”) is calculated. This will be discussed in details during the practical exercises.

19. Examples of tests

20. Tests for the arythmetic mean(s)

21. Tests for proportions

22. Tests for variances

23. Goodness-of-fit tests

24. Thank you!