# That is from the sample of data there is evidence

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That is, from the sample of data, there is evidence that the two population means are the same.

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Econ 325 – Chapter 8 7 Chapter 8.2 More Confidence Intervals for the Difference Between Two Population Means Another problem of interest is to compare the population means of two independent samples. Consider two independent random samples from normal populations: the first sample has x n observations from a population with mean X μ . An estimator of the population mean is the sample mean X . the second sample has y n observations from a population with mean Y μ . 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 2 σ . Econ 325 – Chapter 8 8 The difference between the two sample means Y X - is a normally distributed random variable with mean Y X μ - μ and variance: y 2 x 2 n n ) Y ( Var ) X ( Var ) Y X ( Var σ + σ = + = - From the numeric data set, the calculated sample means and variances are: x , y and 2 x s , 2 y s . An estimate of the population variance 2 σ is needed. A method is to pool (or combine) the data from the two samples and calculate the pooled sample variance : ) 2 n n ( s ) 1 n ( s ) 1 n ( s y x 2 y y 2 x x 2 - + - + - = The degrees of freedom associated with the variance calculation is ) 2 n n ( y x - + .
Econ 325 – Chapter 8 9 With this design, a ) 1 ( α - 100 % confidence interval estimate for the difference in population means ( Y X μ - μ ) is given by: y 2 x 2 c n s n s t ) y x ( + ± - where c t is the critical value from the t-distribution with ) 2 n n ( y x - + degrees of freedom such that: 2 ) t t ( P c ) 2 n n ( y x α = > - + Econ 325 – Chapter 8 10 Example: A stock market data base contains daily closing prices for the company Johnson & Johnson for the year 1999. For the first six months (January to June) observations are recorded for 124 trading days. For the period July to December observations are available for 128 trading days.

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