7.5 Estimating a Population Variance

# 7.5 Estimating a Population Variance - 7.5 Estimating a...

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7.5 Estimating a Population Variance We have seen how confidence intervals can be used to estimate the unknown value of a population mean or a proportion. We used the normal and student t distributions for developing these estimates. However, the variability of a population is also important. As we have learned, less variability is almost always better. We use the chi-square distribution (pronounce as kigh-square) to construct the confidence intervals (estimates) of variances or standard deviations. First we n eed to become acquainted with the χ 2 distribution. χ 2 (chi-square) distribution Suppose we take a random sample of size n from a normal population with mean µ and standard deviation σ. Then the sample statistic follows a χ 2 distribution with n-1 degrees of freedom, where s 2 represents the sample variance. Properties of the χ 2 (chi-square) distribution The total area under χ 2 curve equals 1. The value of the χ 2 random variable is never negative, so the χ 2 curve starts at 0. However, it extends indefinitely to the right, with no upper bound.

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7.5 Estimating a Population Variance - 7.5 Estimating a...

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