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Probability and Samples: The Distribution of Sample Means Chapter 7

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Chapter Overview Samples and Sampling Error The Distribution of Sample Means Probability and the Distribution of Sample Means Computations

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Q? What is the purpose of obtaining a sample? A. To provide a description of a population

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What happens when the sample mean differs from population mean? Sampling Error: The discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. 2 separate samples from the same population will probably differ. –different individual –different scores –different sample means

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Predicting the characteristics of a sample Distribution of Sample Means: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population

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Distribution of sample means are statistics, not single scores. Sampling distribution: a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

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Example 7.1

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Let’s construct a distribution of sample means What do we need to know –Population parameters (scores) 2,4,6,8 –Specify an (n) –Examine all possible samples

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Figure 7.2

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Characteristics of sample means Sample means tend to pile up around the population mean The distribution of sample means is approximately normal in shape. The distribution of sample means can be used to answer probability questions about sample means

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What do we use when we have a large n and do not want to calculate all of the possible samples ?

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Central Limit Theorem CLT: For any population with mean of and a standard deviation , the distribution of sample means for sample size n will approach a normal distribution with a mean of and a standard deviation of /n (square root of n) as n approaches infinity.

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CLT: Facts Describes the distribution of two sample of sample means for any population, no matter what shape, mean, or standard deviation. The distribution of sample means “approaches” a normal distribution by the time the size reaches n= 30.

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Central Limit Theorem Cont’d Distribution of sample means tends to be a normal distribution particularly if one of the following is true: –The population from which the sample is drawn is normal. –The number of scores (n) in each sample is relatively large (n>30)

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Expected value of X Sample means should be close to the population mean aka the expected value of x Expected value of X: the mean of the distribution of sample means will be equal to (the population mean)

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Standard Error of X Notation: x = standard distance between x and The standard deviation of the distribution of sample means. Measures the standard amount of difference one should expect between X and simply due to chance

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Magnitude of the Standard error is determined by The size of the sample The standard deviation of the population from which the sample is selected Law of large numbers: the > n, the more probable the sample mean will be close to the population mean.

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Learning Check pg 151 1)A population of scores is normal with =80 and =20 a)Describe the distribution of sample means for samples of size n=16 selected from this population. (Describe shape, central tendency, and variability, for the distribution) b)How would the distribution of sample means be changed if the sample size were n=100 instead of n=16.

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2) As sample size increases, the value of the standard error also increases? (True or False) 3)Under what circumstances will the distribution of sample means be a normal shaped distribution?

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Learning Check 7.2 pg 152 SAT scores with a normal distribution with a =500 and =100 In a random sample of n=25 students, what is the probability that the sample mean would be greater than 540?

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Figure 7.3 A distribution of sample means Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

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Z-scores for Sample Means Z-scores describe the position of any specific sample w/in the distribution The z-score for each distribution can be calculated using: z=X- x

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General Concepts Standard error: samples will not provide perfectly accurate representations of the population Standard error provides a method for defining and and measuring sampling error.

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Figure 7.6 The structure of research study Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

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Figure 7.8 Showing standard error in a graph Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

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