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Elementary statistics for foresters. Lecture 5 Socrates/Erasmus Program @ WAU Spring semester 2005/2006. Statistical tests. Statistical tests. Why using tests? Statistical hypotheses Errors in tests Test of significance Examples of tests. Why do we use tests?. We work with samples
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Elementary statistics for foresters Lecture 5 Socrates/Erasmus Program @ WAU Spring semester 2005/2006
Statistical tests • Why using tests? • Statistical hypotheses • Errors in tests • Test of significance • Examples of tests
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...
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
Parametric hypotheses • They are usually written as a short equation, e.g. μ = 44 μ1 = μ2 σ1 = σ2
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
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.
Statistical hypotheses H0: μ = 44 H0: μ1 = μ2 H0: the distribution of the „x" variable follows the normal distribution
Statistical hypotheses H1: μ ≠ 44 H1: μ1 ≠ μ2 H1: the distribution of the „x" variable doesn’t follow the normal distribution
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: • H0 is true, test accepts the hypothesis • H0 is true, test rejects the hypothesis • H0 is false, test accepts the hypothesis • H0 is false, test rejects the hypothesis
Errors in test • In two cases we have a bad scenario: • H0 is true, test rejects the hypothesis • H0 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:
Hypothesis / decision Accept Reject true OK Type I error / error of the 1st kind false Type II error / error of the 2nd kind OK Errors in tests
Hypothesis / decision Accept Reject true OK alpha error false beta error OK Errors in tests
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)
Test of significance scheme • formulate H0 and H1, • 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 rejectedor state, that "we can't reject the null hypothesis for a given significance level α”
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.
