Revising the Expected Value and the Variance-ECO6416

Revising the Expected Value and the Variance-ECO6416 -...

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Revising the Expected Value and the Variance Averaging Variances: What is the mean variance of k variances without regard to differences in their sample sizes? The answer is simply: Average of Variances = [ Σ S i 2 ] / k However, what is the variance of all k groups combined? The answer must consider the sample size n i of the ith group: Combined Group Variance = Σ n i [S i 2 + d i 2 ]/N, where d i = mean i - grand mean, and N = Σ n i , for all i = 1, 2, . ., k. Notice that the above formula allows us to split up the total variance into its two component parts. This splitting process permits us to determine the extent to which the overall variation is inflated by the difference between group means. What the variation would be if all groups had the same mean? ANOVA is a well-known application of this concept where the equality of several means is tested.
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Unformatted text preview: Subjective Mean and Variance: In many applications, we saw how to make decisions based on objective data; however, an informative decision-maker might be able to combine his/her subjective input and the two sources of information. Application: Suppose the following information is available from two independent sources: Revising the Expected Value and the Variance Estimate Source Expected value Variance Sales manager μ 1 = 110 σ 1 2 = 100 Market survey μ 2 = 70 σ 2 2 = 49 The combined expected value is: [ μ 1 / σ 1 2 + μ 2 / σ 2 2 ] / [1/ σ 1 2 + 1/ σ 2 2 ] The combined variance is: 2 / [1/ σ 1 2 + 1/ σ 2 2 ] For our application, using the above tabular information, the combined estimate of expected sales is 83.15 units with combined variance of 65.77....
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