7_1_Handout - 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, σ/ n ) 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 s/ n 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 SE x = s/ n one-sample z statistic (6.2) = ( x 0 ) / (σ/ n ) basis for inference about μ when σ is known x distributed normally (or approximately normally)- used Table A when we substitute s/ n for σ/ n , 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 = ( x 0 ) / (s/ n ) has the t distribution with n-1 degrees of freedom . •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
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Why degrees of freedom = n-1? Recall CH 1. If we know x and the values of n-1 of the observations, we can figure out the value of the last observation and, hence the standard
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7_1_Handout - 7.1 Inference for the Mean of a Population...

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