lesson_10.4.1_version_1.6_student

In module 9 we discovered a very important fact

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Unformatted text preview: al or the sample sizes are greater than 30. In order to construct confidence intervals and test claims about the difference of two population means, we first must understand the distribution of differences between sample means. In Module 9, we discovered a very important fact regarding differences of sample proportions: as long as two sampling distributions of sample proportions are approximately normal, then the sampling distributions of all differences between these sample proportions are normal as well. This fact turns out to be true for sample means as well. Whenever two sampling distributions of sample means are approximately normal (this occurs when ! > 30 or the population they are sampled from is normal), then the sampling distribution of their differences is approximately normal as well. To assure this, we need only require that each sample size is at least 30 in size or that the two populations from which we are sampling are normal. Also in Module 9 we noted that the mean of all differences is equal to the difference of the respective means. This idea applies here as well, as we consider the sampling distribution of all differences between sample means, !! − !! . If the sample means have corresponding population means !! and !! , then the mean of all differences between sample means is the difference of the population means: Mean of differences: !! − !! The standard error of the differences in sample means is given by: Standard Error of differences: ! !! !! !! + !! ! Recognizing that the population standard deviations will likely be unknown, we estimate the standard error using sample standard deviations. Estimated Standard Error: ! !! !! !! + !! !...
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