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Sampling Distribution of Means: Basic Theorems. is an unbiased estimate of . Consider N samples each consisting of 2 observations sampled at random from a single population with mean and variance . is the i-th
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Sampling Distribution of Means: Basic Theorems • is an unbiased estimate of . • Consider N samples each consisting of 2 observations sampled at random • from a single population with mean and variance . is the i-th • observation in the j-th sample. T.j is the sum (total) of the 2 observations in • the j-th sample. Observation 1 2 T 1 X11 X21 T.1 2 X12 X22 T.2 . . . . . . . . j X1j X2j T.j . . . . . . . . N X1N X2N T.N Sample Or, more generally, for samples of n observations: As N approaches infinity the distribution of the i-th observation over repeated random samples approaches the distribution of the population from which the i-th observation was drawn. If all Xiare sampled from the same population, then is a constant for all i, . Q.E.D.
2. The variance of the sampling distribution of means for random and independent samples of size n is given by the variance of the population from which the samples were divided by the sample size: Note that For samples of size 2: If X1j is independent of , then and Or more generally, if all Xij are independent, for random samples of n observations: As N approaches infinity the distribution of the i-th observation over repeated random samples approaches the distribution of the population from which the i-th observations was drawn. If all Xij are sampled from the same population, then is a constant for all i, .
is an unbiased estimate of . • Consider the following data matrix. • Note that • To obtain the mean squared deviation of each observation in the data matrix • about the grand mean of all observations, we may proceed as follows: • As N approaches infinity the distribution of all N observations in the data • matrix approaches the distribution of the population from which each • was sampled. Observation 1 2 ….. i …… n ……….M 1 X11 X21 Xi1 Xn1 M.1 2 X12 X22 Xi2 Xn2 M.2 . . . . . . . . j X1j X2j Xij Xnj M.j . . . . . . . . N X1N X2N XiN XnN M.N Sample
3. continued 4. From Theorems 2 and 3it follows that: Is an unbiased estimate of (with n-1 degrees of freedom). Note that:
