Final SummaryThe Central Limit TheoremWe have examined in detail three components of the central limit theorem -- successive sampling, increasing sample size, and different populations. Let's review what we have learned from each and put them together into a final statement. Remember that the central limit theorem applies only to the mean and not to other statistics. Show
General Procedure
Successive Sampling
Increasing Sample Size
Population Distributions
Central Limit TheoremThe central limit theorem states that when an infinite number of successive random samplesare taken from a population, the distribution of sample means calculated for each sample will become approximately normally distributed with mean � and standard deviation s/�N ( ~N(�,s/�N)) as the sample size (N) becomes larger, irrespective of the shape of the population distribution. Hypothesis TestsHow does the central limit theorem help us when we are testing hypotheses about sample means? Even if we do not know the distribution of scores in the original population, we know that the sampling distribution of the means will be approximately normally distributedwith mean� and standard deviation s/�N, if the sample is relatively large. Knowing the properties of the sampling distribution allows us to continue with the test, even if we don't know what the population distribution looks like.
What happens to standard deviation when sample size increases?Thus as the sample size increases, the standard deviation of the means decreases; and as the sample size decreases, the standard deviation of the sample means increases.
What happens to standard deviation when sample size is doubled?If every term is doubled, the distance between each term and the mean doubles, BUT also the distance between each term doubles and thus standard deviation increases.
What happens to the standard deviation of P as the sample size increases if the sample size is increased by a factor of 4 What happens to the standard deviation of P?If the sample size is increased by a factor of 4, what happens to the standard deviation of p-hat? If n is increased by a factor of four, the standard deviation is HALVED because the square root of four is two so in essence you are doubling the number x is divided by.
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