PSYCH 240 1st Edition Lecture 12Section 7: Central Limit TheoremDistribution of Means-Tests we will learn about are based on means of samples that have multiple scores-To figure out the evidence for the alternative is specific enough to meet out standards, we will have to know how typical a sample mean is if the null hypothesisis true-We will have to form a distribution of means to use as the comparison distribution-μM: mean of distribution-σM: standard deviation of distribution of means. Also called the "standard error of the mean"-The central limit tells us how to derive the shape, mean, and variability of the distribution of means given a distribution of scores and a sample size Central Limit Theorem-Shape: usually normal (when some assumptions are met)-Mean: the distribution of means has the same mean as the distribution of scores μM = μ-Variability:oThe variance of the distribution of means is the variance o the distribution of scores divided by the sample sizeoThe standard deviation of the distribution of means is the standard deviation of the score--The sampling distribution gets less variable as sample size increases-The distribution of means will be normal if the distribution of scores is normal regardless of the sample size-If the distribution of scores is any other shape besides normal, then the distribution of means will still be very close to normal if the sample size if 30 or greater. (the shape can be far from normal for smaller sample sizes) Why Does the Distribution of Means Become Less Variable with Higher N?-To get a sample mean that is far away from the population mean, you must draw scores that are consistently far from the mean in one directionoThis becomes less likely with larger sample sizes-If you take a big sample, extreme scores from a skewed distribution of scores will by moderated by less extreme scores in your sampleoAny skew in the distribution of scores becomes "softened" in the distribution of meansoHaving more scores in the sample moderate the extreme scores provides better protection against the
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