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# Sampling Distributions and the t-Distribution Study Guide (page 4)

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Updated on Apr 22, 2013

## Standard Errors

Throughout this lesson, we have talked about the "sample standard deviation of the sampling distribution of the sample mean. "We have done this to emphasize that we are trying to estimate the standard deviation, not of the original population, but of the conceptual population of the sample means that could be derived from repeatedly taking random samples of size n and finding for each. Because the variability in , , and other estimators is so important, the term standard error is used to represent this idea. That is, the standard error of a statistic is the estimated standard deviation of the statistic. Thus, the "sample standard deviation of the sampling distribution of the sample mean" may be simply stated as the "standard error of ".The standard error of the sample proportion is For a sample mean, the standard deviation of is if σ, is known so there would be no need to estimate that quantity using the standard error.However, in practice, one rarely, if ever, knows σ, so this case will not be considered further. If the mean is unknown, the standard deviation is generally unknown so the sample standard deviation is used to estimate it. Thus, when σ is unknown, the standard error of is. Notice that as the sample size increases, the standard error decreases.

## Sampling Distributions and the t-Distribution Study Guide In Short

Each sample produces a sample mean. If we selected another sample, the observations in the sample, and thus the value of the sample mean, are likely to change. When sampling from a normal distribution, the sampling distribution of the sample mean is also normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of n. The sample mean can be standardized using is if the population standard deviation is known and . Because the sample standard deviation of the sampling distribution of the sample mean is frequently used, it is simply referred to as the standard .

Find practice problems and solutions for these concepts at Sampling Distributions and the t-Distribution Practice Exercises.

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