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# day22 - So far we have done inference for a population mean...

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So far we have done inference for a population mean and a population pro- portion, each for a single population. Now, we will do inference on means from two populations. Our goal is to compare the two populations. Inferences about the Differences between Two Population Means: σ 1 and σ 2 Known Example : During the 2003 season, Major League Baseball took steps to speed up the play of baseball games in order to maintain fan interest. For a sample of 60 games played during the summer of 2002 and a sample of 50 games played during the summer of 2003, the sample mean duration of the games was computed. For games in 2002, the sample mean was 172 minutes and for games in 2003, the sample mean was 166 minutes. Assume that σ 1 and σ 2 are both 12 minutes. Notation : Population 1: Population 2: μ 1 = μ 2 = σ 1 = 1

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σ 2 = ¯ x 1 = ¯ x 2 = n 1 = n 2 = To compare the two populations we will use the parameter: μ 1 - μ 2 We can estimate this parameter using the sample means: ¯ x 1 - ¯ x 2 Example : What is an estimate of the difference in average game duration between 2003 and 2002? 2
In order to do inference on μ 1 - μ 2 (i.e. confidence intervals and hypothesis tests), we also need the standard error of ¯ x 1 - ¯ x 2 . This standard error is σ ¯ x 1 - ¯ x 2 = v u u t σ 2 1 n 1 + σ 2 2 n 2 We will assume that we know the value of σ 1 and σ 2 . Confidence Interval The interval estimate of the difference between two population means with σ 1 and σ 2 and known and a confidence coefficient of (1 - α ) is x 1 - ¯ x 2 ) ± z α/ 2 v u u t σ 2 1 n 1 + σ 2 2 n 2 Example :

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