Chapter 7- Probability and Samples-The Distribution of Sample Means PDF

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Chapter 7: Probability and Samples—The Distribution of Sample Means  Sampling Error- natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter  Distribution of Sample Means- collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population o Characteristics: 1) Sample means should pile up around population mean (samples not expected to be perfect but representative of population) 2) Pile of sample means should form a normal-shaped distribution (most samples should have means close to µ ad frequencies should taper off as distance between M and µ increases 3) Larger sample size means the closer the sample means should be to the population mean (µ) (logically, a larger sample should be more representative) o Central Limit Theorem: 1) Describes distribution of sample means for any population (regardless of shape, mean, or standard deviation) 2) Distribution of sample means “approaches” a normal distribution very rapidly 3) Describes distribution of sample means by identifying three basic characteristics of any distribution—shape, central tendency, and variability o Shape of Distribution of Sample Means (perfectly normal distribution if…) 1) Population from which samples are selected is a normal distribution 2) Number of scores (n) in each sample is relatively large (around 30 or more) o Expected Value of M- mean of distribution of sample means is equal to mean of population of scores, µ 1) Standard Error of M- (σM) standard deviation of distribution of sample means a) Describes distribution of sample means, provides a measure of how much difference is expected from one sample to another— when it’s small, all sample means are close together and have similar values, but when it’s large, the sample means are scattered over a wide range and there are big differences from one sample to another b) Measures how well an individual sample mean represents entire distribution, as well as measure of how much distance to expect between a sample mean (M) and the population mean (µ) o Law of Large Numbers- states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean 1) When n = 1, σM = σ (standard error = standard deviation) 2) As sample size increases(n) increases, the size of the standard error decreases (larger samples are more accurate) 3) Standard Error = σM = (σ/√n) = √(σ2/n) o z = (M – µ)/ σM

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Sampling Distribution- distribution of statistics obtained by selecting all the possible samples of a specific size from a population...