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Inference about the Difference Between Two Means

# Inference about the Difference Between Two Means -...

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1 Inference about the Difference Between Two Means The objective of many studies is to compare two means in addition to estimating individual means. Suppose there would are two populations, one with mean μ 1 and the other with mean μ 2 , and we want to make inference about the difference μ 1 - μ 2 . We might want to test the null hypothesis H 0 1 2 , or construct a confidence interval for the difference μ 1 - μ 2 . Which statistical method you use depends on how you obtained the data. Basically, there are two types of samples, independent and paired . Independent Samples: You have a sample of n 1 observations from population 1 and a sample of n 2 observations from population 2. The observations of the first sample are denoted y 11 ,…,y 1n1 , and the observations of the second sample are denoted y 21 ,…,y 2n2 , Paired Samples: You have two samples, but each observation in the first sample is related to an observation in the second sample. The related observations make a pair. The data are denoted (y 11 ,y 21 ),…,(y 1n ,y 2n ), where y 1i and y 2i are two observations that are related from samples 1 and 2, respectively.

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2 Inference about the Difference Between Two Means Comparing means from two independent samples 0 1 2 1 2 : : a H H µ µ µ µ = > Test Statistic: ( ) ( ) 2 2 1 1 2 2 2 1 2 1 2 2 1 2 1 2 1 1 , 2 1 1 2 p p n s n s y y t s n n s n n d f n n + = = + + = + where 1 y and s 1 2 are the mean and variance from sample 1, and 2 y and s 2 2 are the mean and variance from sample 2. The sample variance s 1 2 and s 2 2 have n 1 – 1 and n 2 – 1 degrees of freedom, respectively.
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