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# Mt139C8 - Sampling Distribution The Sampling Distribution...

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MT139~9.PPT 1 Sampling Distribution The Sampling Distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size drawn from the same population. Sampling Distribution Mean and Standard Deviation First consider x. Suppose this is the mean of an SRS of size n drawn from a population with mean μ and standard deviation σ . The mean of the sampling distribution of x is : The standard deviation of the sampling distribution of x (also called the “standard error of the mean”) is: μ μ = x n x σ σ =

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MT139~9.PPT 2 Statistical Confidence The 68-95-99.7 rule says that in 95% of all samples, the mean score x for the sample will be within two standard errors of the population mean score μ . μ 95% x 2 σ μ - x 2 σ μ + n x σ σ = Sampling Distribution of x
MT139~9.PPT 3 Statistical Confidence (Continued) If x is within two standard errors of the unknown μ , then μ is within two standard errors of the observed x. μ x x 2 σ + x 2 σ - x 2 σ + x 2 σ -

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MT139~9.PPT 4 Statistical Confidence (Continued) So in 95% of all samples, the unknown μ lies within two standard errors of the observed sample mean x. μ x 2 x σ - x 2 x σ + Sampling Distribution of x x
MT139~9.PPT 5 Confidence Interval A level C Confidence Interval for a parameter (the population mean μ , for example) is an interval about a sample estimate (the sample mean x) that will capture the unknown parameter C% of the time in repeated sampling and estimation. Example: μ x x x x x

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MT139~9.PPT 6 Confidence Interval A level C Confidence Interval for a parameter has two parts: 1. An interval calculated from the data, usually of the form: 2. A Confidence Level C, which gives the probability that the interval will capture the true but unknown parameter in repeated samples. Example: Height is recorded from 100 individuals and the sample mean x is found to be 67.89 inches. The standard deviation is known to be 5.20 inches. Construct an approximate 95% confidence interval for the average height in the population. Error of Margin Estimate ± 04 . 1 89 . 67 100 20 . 5 2 89 . 67 n 2 x ± ± ± σ
MT139~9.PPT 7 Confidence Interval for the Population Mean The reasoning we used to find an approximate 95% Confidence Interval for the population mean μ applies to any Confidence Level C.

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Mt139C8 - Sampling Distribution The Sampling Distribution...

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