# Anestimateofthepopulationvariance isneeded

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Unformatted text preview: • the second sample has n y observations from a population with mean μ Y . An estimator of the population mean is the sample mean Y . Note that the samples can have different sample sizes. To develop results, assume that the two populations have the same (unknown) variance σ . 2 7 Chapter 9 The difference between the two sample means X − Y is a normally distributed random variable with mean μ X − μ Y and variance: Var( X − Y ) = Var( X ) + Var( Y ) σ2 σ2 = + nx ny From the numeric data set, the calculated sample means and 2 2 variances are: x , y and s x , s y . An estimate of the population variance σ is needed. A method is to pool (or combine) the data from the two samples and calculate the pooled sample variance: 2 s= 2 ( n x − 1) s 2 + ( n y − 1) s 2 x y ( n x + n y − 2) The degrees of freedom associated with the variance calculation is (n x + n y − 2) . 8 Chapter 9 With this design, a 100 (1 − α) % confidence interval estimate for the difference in population means ( μ X − μ Y ) is given by: (x − y ) ± t c s2 s2 + nx ny where t c is the critical value from the t‐distribution with (n x + n y − 2) degrees of freedom such that: 9...
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