Elementary statistics for foresters

1. Lecture 4 Socrates/Erasmus Program @ WAU Spring semester 2006/2007 Elementary statistics for foresters

2. Statistical estimation

4. 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

5. 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

6. 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

7. Estimation – basic concepts • Sample mean • Sample mean distribution • Standard error of the sample mean • Significance level and confidence level

9. Estimation – an example • Sample data (population): density of wood • Arithmetic mean: 498,76 kg/m3 • Standard deviation: 52,77 kg/m3

11. 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

13. Estimation – an example 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

14. 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

15. 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

16. Estimation – an example Estimation – an example • This probability is referred to as confidence level (beta)‏ • 1 – beta = alpha <- significance level (the probability of error)‏

17. Sample size determination • Closely connected to the estimation process • The equation derived directly from the confidence interval formulae