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Understanding Statistical Estimation Techniques for Forest Management

This lecture covers essential concepts in statistical estimation, focusing on inferential statistics tailored for forestry applications. Key topics include unbiased estimators, sample means, confidence intervals, and the significance of sample size. The lecture emphasizes practical examples, such as estimating wood density based on samples drawn from a population. Students will explore the relationships between sample size, standard error, and confidence levels, equipping them with the necessary tools for making informed decisions in forestry through statistical analyses.

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Understanding Statistical Estimation Techniques for Forest Management

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

  2. Statistical estimation

  3. 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

  4. 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

  5. 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

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

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

  8. 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

  9. 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

  10. 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

  11. 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

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

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

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