# The standard deviation of a sampling distribution

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- the standard deviation of a sampling distribution
Notation: xModules 9 & 10 (Combined) Course Notes Page 3
Formula: x **Sampling distribution of the mean- the distribution of sample means over repeated sampling from one population Goal: -estimate a population mean givensample mean(s) -as increase sample size (or the number of samples), approximate a normal curve 2.Central Limit Theorem **Central Limit Theorem- Given a population with mean and variance 2, the sampling distribution ofthe mean will have a mean equal to and a variance equal to 2/ N. The distribution will approach the normal distribution as N, the sample size, increases. -the rateat which the sampling distribution of the mean approaches normal is a function of the shape of the parent population: a)-if the population is normal, the sampling distribution of the mean will be normal regardless of Nb)-if the population is symmetric but non-normal, the sampling distribution of the mean will be nearly normal even for quite small sample sizes c)-if the population is markedly skewed, sample sizes of 30 or more are required before the means approximate a normal distribution Examples: Sampling Distributions1.Use your z-tables to find the critical z-values: a.= .05 2-tails b.= .05 1-tail c.= .01 2-tails d.= .01 1-tail 2.Use your t-tables to find the critical t-values: 3.The national average for the verbal section of the Graduate Record Exam (GRE) is 500 with a standard deviation of 100. A researcher uses a sampling distribution made up of samples of 100. a.According to the Central Limit Theorem, what is the mean of the sampling distribution of means? b.According to the Central Limit Theorem, what is the standard error of the mean? Modules 9 & 10 (Combined) Course Notes Page 4 n
2a. tcritical = 2.064 (remember degrees of freedom are 24) 2b. tcritical= +2.492 or –2.492 (depending on the direction of the hypotheses) 2c. tcritical = 2.756 (remember degrees of freedom are 29) 2d. tcritical = +1.699 or –1.699 (depending on the direction of the hypotheses) 3a. 500 3b. 10 Hypothesis Testing (steps 4 to 8): 1) Z-Score Hypothesis Testing with Single X-ValueX z-score formula z***This formula is used when both and are known
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