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Estimation in Sampling!? Chapter 7 – Statistical Problem Solving in Geography

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Goals Basis Concepts in Estimation Point Estimation and Interval Estimation Sampling Distribution of a Statistic Central Limit Theorem Confidence Intervals and Estimation Standard Normal and Z-Scores General Procedure for Constructing a Confidence Interval Geographic Examples of Confidence Intervals Sample Size Selection Mean, Total and Proportion in Sample Size Selection

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Points in Estimation

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Intervals in Estimation Interval Estimation Due to the nature of uncertainty it is unlikely that a sample statistic will equal a population parameter. Used to determine the distance that a sample statistic is from a population parameter. Interval estimation uses a confidence interval to establish the likelihood that a sample statistic is within an interval or range from the population parameter. Confidence Interval: Represents level of precision associated with the population estimate. Width is determined by 1) sample size; 2) amount of variability in the population’; and 3) the probability level or level of confidence selected for the problem.

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Sampling Distribution of a Statistic A single sample of size n will lead to a distribution curve which could be any of the curves that we have discussed. Examples are Poisson, Uniform, Normal, etc. This single sample will produce a sample mean and standard deviation. Sampling Distribution of Sample Means: If you take multiple, similar-sized independent samples from a population the set of sample means can be graphed. The red curve is the Sampling Distribution of Sample Means The black curve represents the frequency distribution of values within the population

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Central Limit Theorem Given the effect of randomness in drawing samples, some sample means will fall above the population mean and some below. Provided they are independent samples the mean of the all of the sample means will be the population mean. The distribution of sample means will also be normal and centered on the population mean regardless of the distribution of the population provided that the sample is larger than 30. When the sample size (n) is large, the sample mean(s) will be closer to the population mean.

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Central Limit Theorem

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Confidence Intervals and Estimation

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Z-Scores and Confidence Intervals In order to establish a confidence interval we must determine a z-score. This can be done by looking at a table to see z-scores of common confidence intervals! More information on z-scores can be found at this website.website

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Using Interval Estimates

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Constructing a Confidence Interval

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What Level of Confidence?

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The Real World: Unknown Population Standard Deviation

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What if Sample Size is Small?

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The T-Table

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But! To Calculate a Confidence Interval…. Equation used depends on two factors The equation used for a confidence interval depends on the sample type (random, systematic, stratified, etc.) Different population parameters require different confidence interval equations.

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Random or Systematic Samples

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Random or Systematic Samples Continue….

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Stratified Samples

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Sample Size Selection

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Chapter VII Ending

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