Topic4_StatisticalInference_ECO220_handout (2).pdf - Topic 4 Statistical Inference I Sampling Distributions Hypothesis Testing Confidence Intervals

Topic4_StatisticalInference_ECO220_handout (2).pdf - Topic...

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Topic 4: Statistical Inference I: Sampling Distributions, Hypothesis Testing & Confidence Intervals ECO220Y5Y: Quantitative Methods in Economics Dr. Nick Zammit University of Toronto Department of Economics Room KN3272 n.zammit @ utoronto.ca November 25, 2019 Dr. Nick Zammit (UofT) Topic 4 November 25, 2019 1 / 31 Road Map – Statistical Inference I: Sampling Distributions, Hypothesis Testing & Confidence Intervals Key Concepts: 1 Definitions (Hypothesis Testing, Confidence Intervals, Margin of Error, p-values) 2 Tables/Plots (Normal Critical Values, Student-t Critical Values, χ 2 -Critical Values, Confidence Regions vs. Rejection Regions) 3 Test Statistics/Intervals (One Proportion Intervals, One Mean Intervals, Difference of Proportions/Means) 4 Ideas (Central Limit Theorem, Type I & Type II Errors, Level of Confidence/Significance, Power of Tests, Setting Sample Size) Dr. Nick Zammit (UofT) Topic 4 November 25, 2019 2 / 31 Notes Notes
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What are Sampling Distributions? Sampling Distribution The distribution of proportions or means over many independent samples from the same population A distribution of sample distributions Sampling Error The sampling variability from one sample to another The larger the sample size the smaller the sampling error Dr. Nick Zammit (UofT) Topic 4 November 25, 2019 3 / 31 Empirical Laws Central Limit Theorem Whatever the distribution of X , as the number of terms in the sum becomes large, the distribution of X tends to a normal distribution Applies regardless of whether the underlying distribution is continuous and symmetric like the uniform distribution, continuous and asymmetric like the chi- squared distribution, or even discrete such as the Binomial distribution. CLT implies random samples can be taken from a population and the distribution of sample statistics is normally distributed Dr. Nick Zammit (UofT) Topic 4 November 25, 2019 4 / 31 Notes Notes
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The Central Limit Theorem Example 0 0 5 5 10 10 15 15 20 20 Percent Percent 0 1 1 2 2 3 3 4 4 5 6 6 6Dice Roll Dice Roll Dice Roll Random Dice Rolls (500 Obs) Random Dice Rolls (500 Obs) 0 5 10 15 20 Percent 0 1 2 3 4 5 6 Dice Roll Random Dice Rolls (500 Obs) 0 5 10 15 20 Percent 0 1 2 3 4 5 6 Dice Roll Random Dice Rolls (500 Obs) 0 5 10 15 Percent 3.3 3.3 3.4 3.4 3.5 3.5 3.6 3.6 3.7 3.7 3.8 3.8 E(X) E(X) Mean Dice Rolls (500 obs, 500 means) Mean Dice Rolls (500 obs, 500 means) Dr. Nick Zammit (UofT) Topic 4 November 25, 2019 5 / 31 Sampling Distributions vs. Samples Sampling Distribution Distribution of estimators (‘distribution of distributions’) Becomes normally distributed according to CLT Samples Empirical distributions drawn from underlying population/theoretical distribution Becomes distributed as underlying population/theoretical distribution based on LLN Dr. Nick Zammit (UofT) Topic 4 November 25, 2019 6 / 31 Notes Notes
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Characteristics of Sampling Distributions Sampling Distributions for Proportions Given several assumptions/conditions are satisfied then the sampling distribution of ˆ p is modelled by a Normal distribution with μ p ) = p and SD p ) = q pq n Sampling Distributions for Means Given several assumptions/conditions are satisfied then the sampling distribution of ¯ x
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