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UASTAT141Ch26 - Ch 26 Comparing Counts Notation k = of...

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Ch. 26 - Comparing Counts Notation: k = # of categories of a qualitative variable p i = true proportion of category i ; i = 1,…, k (Note: = k i i p 1 = 1) A random sample of size n will provide sample statistics of “observed counts”. These values can compare against “expected counts” of np i for each category. Consequently, an H 0 can collectively test the validity of each p i . How? Def’n: The “goodness-of-fit” test uses the chi-square statistic , χ 2 , is computed by = cells Exp Exp Obs 2 2 ) ( χ where Obs = “observed count”, Exp = “expected count”, and you sum over all categories. Sizeable differences between Obs and Exp of specific categories lead to large values of χ 2 and subsequent rejection of H 0 . For formal rejection/non-rejection, we need a formal test. Aside: The chi-squared distribution has the following properties: - like the t -distribution, it has only one parameter, df , that can take on any positive integer value. - skewed to the right for small df but becomes more symmetric as df increases. - curve where all areas correspond to nonnegative values. - values denoted by χ 2 When H 0 is correct and n sufficiently large, χ 2 approx. follows a χ 2 -dist’n with df = k – 1. Using this dist’n, the corresponding P -value is the area to the right of χ 2 under the χ 2 k -1 curve (all curves found in Appendix Table X).
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