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UA PSY 230 - Probability and Samples

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Lecture 6: Let’s Start Inferential StatsLet’s Do an ExperimentToday’s GoalSlide 4For Example:How do we know how closely our sample represents our population?Sampling ErrorDistribution of Sample MeansSlide 9PredictionsSlide 11Central Limit TheoremSlide 13A little more about the shape and mean of the distribution of sample meansStandard Error of MStandard ErrorMore about Standard ErrorIllustration - Let’s do it!Let’s Do it!Z-scores for sample meansLet’s Do itOne More…Z - Scores and Sample MeansFurther application of Z-testsClass ProblemIn the LiteratureSlide 27Homework: Chapter 7Lecture 6: Let’s Start Inferential StatsProbability and Samples:The Distribution of Sample MeansLet’s Do an ExperimentImagine a jar filled with marbles. 2/3 of the marbles are one color and the remaining 1/3 is a different color.–Sample 1: N = 5; red = 4, white = 1 (80% red).–Sample 2: N = 20; red = 12, white = 8 (60% red).Which sample are you more confident came from a population of 2/3 red and 1/3 white balls? Why?Tversky and Kahneman (1974) found that most people tend to focus on the sample proportion than the sample size, but when asked how many balls they would like to select to make their decision people preferred the opportunity to select 20 v. 5.Today’s GoalFirst from z-scores and probabilities we KNOW:–how scores relate to each other in a distribution–how an individual score relates to its population–Where scores fit into their distributions (probabilities)•Are they representation •Are they extreme? To understand the relationship between samples and populationsBut…we only know about samples that are made up of a single individual score–Most researcher take much larger samples•E.g. 100 specimen, 30 dogs, 50 math scoresPopulationsMembers must share at least 1 traitThe more traits,–the lower the ability to generalize–the smaller the population sizeSamplesGreater the n, the more accurate the parameter estimate (more chances you’ve got to accurately represent the population)Representative Sample: sample which possess all the defining characteristics of the population from which it was drawnFor Example:Say we want to learn about college students at the UA. We randomly choose 30 students at the UA.–We chose our sample randomly it should be pretty representative of the population of students at the UA, but we may be missing some segments of the population (e.g. what if by chance our sample includes no Christians, or any international students?)–Any corresponding stats we compute for the sample will also not be identical to the corresponding parameters–What if we choose another random sample?How do we know how closely our sample represents our population?Z-scores: where a single score lies in its population AND where a sample mean lies in its population.Samples give use an incomplete and often inaccurate picture of our population, so we keep track of sampling error:–Sampling error: the discrepancy or amount of error between a sample statistic and its population parameter.Sampling ErrorSamples never precisely reflect the populationThe difference between the parameter & statistic is sampling error( - M)Sampling error is expected & normalp (+ sampling error) = p (- sampling error)–Some samples overestimate and some underestimate–This error should be randomf1 2 3 4 5 6 7 98Distribution of Sample Means 2 samples taken from the same population will probably be different–Different individuals -- Different means–Different scores -- Different standard deviationsGiven that we can take some extremely large # of samples…what pattern might these samples show?Distribution of Sample Means (or sampling distribution) - all the possible random samples of a particular size (n) that can be taken from a populationSo, we can compute probabilities p(particular sample) = # particular sample/all samplesDistribution of Sample MeansSampling distribution - is a distribution of statistics (means of samples).Consider a population of 4 scores: 2, 4, 6, 8f1 2 3 4 5 6 7 98* See also…Box 7.1 in the Book…page 205PredictionsWhat would we expect if we created a distribution of all the possible n = 2 samples of our data set–Sample means won’t always be perfect, but should pile around pop. mean–Should start to form a normal distribution b/c most of the sample means should pile around the pop. Mean, only a few should be extreme–Larger the sample size the closer the sample mean should be to the population mean b/c a larger sample should be more representativef1 2 3 4 5 6 7 98* If we chose samples of n = 2, then we can have a total of 16 different possible samples = 5Sample 1st score 2nd score M12345678910111213141516222244446666888824682468246824682345345645675678f1 2 3 4 5 6 7 98(1) Sample means pile around pop. mean. (they are representative.)(2) Distribution is ~normal (3) We can use this sample distribution to answer probability questions.e.g. What is the probability of obtaining a sample less than 3?P (M < 3) = 1/16 or 0.06Central Limit TheoremNot reasonable to take all the possible samples in a pop. Usually we just take one.Central Limit Theorem - general characteristics about the sample mean–For any population with mean  and standard deviation , the distribution of sample means for sample size n will have a mean of  and a standard deviation of /n, and will approach a normal distribution as n approaches infinity.Central Limit TheoremPerks:Describes the distribution of sample means for any population regardless of original shape, mean or standard deviation–Shape–Central tendency–VariabilityImportant mathematical finding:–Sampling distribution of mean has a mean = population mean and variance = population variance/nA little more about the shape and mean of the distribution of sample meansShape:–Normal if the samples come from a population that is normal–Normal if the number of scores (n) in each sample is around 30 or more.–What does this mean for researchMean:–Average of all sample means = population mean.–This mean value is call the expected value of M. M (b/c this value will always be equal to , this book will just use  to refer to the mean for both the pop. and the mean for the distribution of sample meansStandard Error of MStandard deviation for a distribution of sample means is called standard error


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UA PSY 230 - Probability and Samples

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