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handout_7_1 - 7.1 Inference for the Mean of a Population...

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7.1 Inference for the Mean of a Population the sampling distribution of x depends on σ when σ is unknown, we must estimate σ even though we are primarily interested in μ the sample standard deviation (s) is used to estimate the population standard deviation (σ) x has N(μ, n σ ) distribution when population has N(μ, σ ) when σ is unknown, we estimate it with the sample standard deviation (s) we estimate the standard deviation of x by n s Standard error When the estimated standard deviation is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean is x SE = n s one-sample z statistic (6.2) = z = - n x σ μ ) ( 0 basis for inference about μ when σ is known x distributed normally (or approximately normally)- used Table A when we substitute n s for n σ , our statistic is not distributed normally the statistic now has a t distribution The t distributions Suppose that an SRS of size n is drawn from a N(μ, σ) population. Then the one-sample t statistic t = - n s x ) ( 0 μ has the t distribution with n-1 degrees of freedom .
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•there is a different t distribution for each sample size (Table D) •a particular t distribution is specified by giving the degrees of freedom •we use t(k) to stand for the t distribution with k degrees of freedom •The density curves of the t(k) distributions are similar in shape to the standard normal curve (unimodal, symmetric about 0, and bell-shaped).
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