The x we calculate is a weighted sum of n random

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Unformatted text preview: out them as a collection n random variables which are identical to the X . The X we calculate is a weighted sum of n random variables, so a random variable itself. Utku Suleymanoglu (UMich) Sampling Distributions 8 / 21 Sampling Distributions Mean of X Remember X = n Xi i =1 n = X1 +X2 +···+Xn . n We can take the expectation of X . X1 + X2 + · · · + Xn ) n 1 1 1 = E ( X1 + X2 + · · · + Xn ) n n n 1 = E (X1 + X2 + · · · + Xn ) n 1 = (E (X1 ) + E (X2 ) + · · · + E (Xn )) n 1 = ( µ + µ + · · · + µ) n 1 = ( n × µ) n =µ µX = E ( X ) = E ( So the expected of the sample mean is the population mean if we do random sampling. Utku Suleymanoglu (UMich) Sampling Distributions 9 / 21 Sampling Distributions A key thing here: We don’t know what µ is. So how on earth the expected value of X equals to an usually unknown thing and we claim the knowledge of this? Notice that it is not some x = µ, it is the expected value of it equals to µ. This result, E (X ) = µ tell us : if we keep on randomly getting samples from this population of interest and building...
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