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Chapter 10

ECO220Y1 Chapter 10 Notes

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University of Toronto St. George
Jennifer Murdock

ECO220Y1 Textbook Notes Chapter 10: Sampling Distributions  Each proportion is based on a different sample of the population.  The proportions vary from sample to sample because the samples comprise different people. 10.1 Modelling Sample Proportions  p = “true proportion”  A simulation can help understand how sample proportions vary due to random sampling.  ̂ = proportion of success  The way in which the proportions vary sample to sample shows how the proportions of real samples would vary. 10.2 The Sampling Distribution for Proportions  Sampling distribution: the distribution of a statistic (proportion) over many independent samples of the same size from the same population. o The Normal approximation can be used provided the sampled values are independent and the sample size is large enough. ( ̂ ( ) √ ̂  Sampling variability: the variability we expect to see from sample to sample.  Using the Normal approximation is valuable because: o The Normal sampling distribution model tells us what the distribution of sample proportions would look like. o Since the Normal model is a mathematical model, we can calculate what fraction of the distribution will be found in any region.  Using the standard Normal table.  The Normal model becomes a better and better representation of the distribution of the sample proportions as the sample size gets bigger.  Two assumptions in the case of the model for the distribution of sample proportions: o Independence: sample values must be independent of each other.  To check the independence assumption, check the following conditions:  Randomization Condition o Sampling method must not have been biased and the data must be representative of the population.  I.e. in an experiment subjects are randomly assigned to treatments.  10% Condition o The size of n must not be > 10% of the population.  Success/Failure Condition o At least 10 successes and 10 failures must occur. o Sample Size Assumption: the sample size, n, must be large enough. 10.3 The Central Limit Theorem – The Fundamental Theorem of Statistics  Proportions summarize categorical variables.  The sampling distribution of almost any mean becomes Normal as the sample size grows. o Observations need to be independent and randomly collected. o The shape of the population distribution does not matter.  Central Limit Theorem (CLT): The CLT states that the sampling distribution model of the sample mean (and proportion) is approximately Normal for large n, regardless of the distribution of the population, as long as the observations are independent. o This is true regardless of the shape of the population distribution. o It is an imaginary distribution since we don’t actually draw all
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