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Unformatted text preview: Chapter 8.1 Confidence Intervals for the Difference Between Two Population Means Let for i = 1, 2, . . . , n be a pair of random variables that each follow the normal distribution with population means and. )Y,X( i i X P Y P This set-up is called matched pairs . A data set is collected with the numeric observations: for i = 1, 2, . . . , n )y,x( i i Example: For n students in Econ 325, is the mid-term grade of student i, and i x is the final grade of student i. i y The interest is to develop a confidence interval for the difference in the population means: Y X P . Econ 325 Chapter 8 1 From the observed sample calculate the differences: for i = 1, 2, . . . , n i i i y x d Obtain the sample mean and variance of the differences as: y x )y x( n 1 d n 1 d n 1i i i n 1i i and xy 2 y 2 x n 1i 2 i 2 d s2 s s )d d( 1 n 1 s . where and are the sample variances from the two variables and is the sample covariance. 2 x s 2 y s xy s Note that the variance of the differences recognizes the covariance between the two variables. Econ 325 Chapter 8 2 A% confidence interval estimate for the difference in population means ( ) 1( D0 100 Y X 0P ) is given by: n s t d d c r where is the critical value from the t-distribution with ( n 1 ) degrees of freedom such that: c t 2 )t t(P c )1n( D " Econ 325 Chapter 8 3 Example: Stock market data for 20 successive business days has been collected. The data set has the observations: ,, . . . , daily percentage returns for a company, and 1 x 2 x n x , , . . . , daily percentage returns for a market portfolio. 1 y 2 y n y On a given day, stock market prices respond to information about the general economy and therefore, it may be expected that the returns for an individual company may be correlated with the overall performance of the market. That is, a non-zero covariance between the x and y observations is realistic. The task of interest is to obtain a confidence interval estimate for the...
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- Spring '10