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

• 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:

Errors in tests

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

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 rejected or 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.

Examples of tests

Tests for the arythmetic mean(s)

Tests for proportions

Tests for variances

Goodness-of-fit tests

Thank you!