Elementary statistics for forestersLecture 5

Socrates/Erasmus Program @ WAUSpring semester 2005/2006

Statistical tests

Statistical testsWhy using tests?Statistical hypothesesErrors in testsTest of significanceExamples of tests

Why do we use tests?We work with samplesWe want to know about populationsSample = uncertaintySo: we need a tool to be able to answer questions about population based on results from the sampleSome examples...

Statistical hypothesesHypothesis: it is a statement about parameters or variable distribution of populationHypothesis refers to a parameter parametric hypothesisHypothesis refers to a distribution non-parametric hypothesis

Parametric hypothesesThey are usually written as a short equation, e.g.

= 441 = 21 = 2

Non-parametric hypothesesUsually written as a sentence, such as e.g.the distribution of the x variable in the population follows the normal distributionsamples were drawn from populations having the same distributions...Used not only exactly for distributions

Statistical hypothesesNull hypothesis a hypothesis being tested during the testing procedureAlternative hypothesis a reserve hypothesis used when the null hypothesis is not trueThese hypotheses can be both: parametric and non-parametric.

Statistical hypothesesH0: = 44H0: 1 = 2H0: the distribution of the x" variable follows the normal distribution

Statistical hypothesesH1: 44H1: 1 2H1: the distribution of the x" variable doesnt follow the normal distribution

Errors in testsThe hypothesis can be: true or falseThe result of the test can be: accept or reject the null hypothesisAll possible cases are:H0 is true, test accepts the hypothesisH0 is true, test rejects the hypothesisH0 is false, test accepts the hypothesisH0 is false, test rejects the hypothesis

Errors in testIn two cases we have a bad scenario:H0 is true, test rejects the hypothesisH0 is false, test accepts the hypothesisIn these cases we have an error in using a statistical testAll cases can be shown in the table:

Errors in tests

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 schemeformulate 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 levelreject the null hypothesis is rejected or state, that "we can't reject the null hypothesis for a given significance level

Test of significance in practiceWhen 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.

Examples of tests

Tests for the arythmetic mean(s)

Tests for proportions

Tests for variances

Goodness-of-fit tests

Thank you!