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Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chapter 6 Hypothesis Tests with Means of Samples
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall The Distribution of Means The distribution of means is –A distribution of the means of a large number of samples of the same size –Each sample randomly selected from the same population of individuals
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Characteristics of the Distribution of Means Mean –Same as the mean of the population of individuals Variance –Variance of the population of individuals divided by the number of individuals in each sample –SD of the distribution of means is the square root of the variance of the distribution of means Shape –Approximately normal if either Each sample as at least 30 observations Underlying distribution of the population of individuals is normal –In general, unimodal, symmetrical, less spread out than the distribution of the population of individuals
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Review of Three Kinds of Distributions Weve considered three kinds of distributions A.The distribution of the population of individuals B.The distribution of a sample C.The distribution of means of samples
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Hypothesis Testing with a Distribution of Means Figuring Z score of a samples mean on the distribution of means –Conceptually the same as figuring a Z score for an individual on a distribution of a population –Subtract the mean of the distribution of means from the samples mean –Then divide by the SD of the distribution of means
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Estimation and Confidence Intervals Point estimate –Estimate of a specific value of a population parameter (e.g., the mean) –The best estimate of the mean of a population is the mean of a sample taken from that population –SD of the distribution of means (called the standard error) is a measure of the accuracy of the point estimate
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Estimation and Confidence Intervals Confidence interval –Interval estimate with a given likelihood of including the population mean –Typical confidence intervals 95% 99% –The larger the interval, the greater ones confidence than the interval contains the true mean
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Steps for Figuring 95% and 99% Confidence Intervals Estimate population –Best estimate is sample mean Figure SD of distribution of means Find Z scores that go with confidence limits you want –For 95%, Z scores are and – For 99%, Z scores are and Convert Z scores to raw scores –Multiply Z score by SD of distribution of means and add to the population mean
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Hypothesis Testing Confidence intervals sometimes used instead of, or alongside, ordinary hypothesis testing –If confidence interval does not include the mean of the null hypothesis distribution, result is statistically significant Rare for researchers to know the characteristics of the population distribution, but when they do, the hypothesis test that is appropriate is a Z test.
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