Elementary statistics for foresters

Lecture 4

Socrates/Erasmus Program @ WAU

Spring semester 2006/2007

Statistical estimation

Estimation

• Inferential statistics• Drawing conclusion about population based

on sample• Drawing conclusion about parameter based

on estimator• Using an estimator to assess the value of the

parameter

Estimator

• Statistics from the sample used to figure out about population parameter

• First of all: unbiased (means: not giving a sistematic error)– E(Tn) = Θ– E(Tn) - Θ = b(Tn) <- bias

• Effective– Having the lowest possible variance

• Other properties

Estimation

• Estimation can be done using two basic techniques:

• Point estimation– Parameter = Estimator

• Confidence interval– Building the interval where we expect the

parameter with a given probability

Estimation – basic concepts

• Sample mean• Sample mean distribution• Standard error of the sample mean• Significance level and confidence level

Estimation – an example

• Sample data (population): density of wood • Arithmetic mean: 498,76 kg/m3• Standard deviation: 52,77 kg/m3

Estimation – an example

• Let's draw 10 000 samples of 10 elements each from our population

• Let's calculate arithmetic mean for each sample

• Mean of means: 498,43 kg/m3 – it's VERY close to the true mean

Estimation – an example

• The histogram of 10 000 means is the normal distribution, so we can use the theory of the normal distribution to arithmetic mean from ANY sample

• Standard deviation of 10 000 means: 16,25 kg/m3 <- it is smaller than the standiard deviation in our population

• Standard deviation of sample means is called STANDARD ERROR

Estimation – an example

Estimation – an example

• Standard error depends on sample size• If sample size = population size: standard

error = 0• If sample size = 1: standard error = standard

deciation of the population• Any other sample size: standard error =

standard deviation of populations / square root of the sample size

Estimation – an example

Estimation – an example

• From the normal distribution theory:– Probability, that the true mean is between

arithmetic mean +/- one standard error = 0,68– Probability, that the true mean is between

arithmetic mean +/- two standard errors = 0,95– Probability, that the true mean is between

arithmetic mean +/- three standard errors = 0,997

Estimation – an example

Estimation – an example

• This probability is referred to as confidence level (beta)

• 1 – beta = alpha <- significance level (the probability of error)

Estimation – an example

Sample size determination

• Closely connected to the estimation process• The equation derived directly from the

confidence interval formulae