By The Mathematical Association of America

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**Extra resources for American Mathematical Monthly, volume 107, number 7, August-September 2000**

**Sample text**

Make the simplifying assumption that the cell counts are large enough to justify an asymptotic argument that replaces multinomial distributions with Gaussian approximations. Then the p-value is approximately equal to the probability that a chi-square random variable with 121 = (12 - 1) X (12 - 1) degrees of freedom is greater than or equal to X 2 • This is "old-style" statistics, in that simplifying assumptions that mayor may not fit the data at hand are needed to render the problem analytically tractable.

47 (2000) 231-243. Don O. Loftsgaarden, Donald C. Rung, and Ann E. 2, Mathematical Association of America, Washington, DC, 1997. -Sept. 2000] STATISTICS AND MATHEMATICS 629 20. David S. Moore, Tests of chi-squared type, in Ralph B. D'Agostino and Michael A. ), Goodness-of-Fit Techniques, Marcel Dekker, New York, 1986, pp. 63-95. 21. Mount Holyoke College, Laboratories in Mathematical Experimentation: A Bridge to Higher Mathematics, Springer-Verlag, New York, 1997. 22. National Council of Teachers of Mathematics, Curriculum and Evaluation Standards For School Mathematics, NCTM, Reston, VA, 1989.

For each k E Z+ let Fk be the following closed set: U Bf-1rn(x Fk = n ). -Sept. 2000] Fk+ 1 and = n k~ 1 Fk = {B eX: 3 k E 0. One can now check that Z+ such that B n Fk A FAMILY OF BOUNDED SETS? IJ is {F: : n E Z+}). 91' there exists k E Z+ such that A n(U~=kv,,) = 0 and so An Fk = 0. IJ, as required. Hu's result leads to a second completely transparent answer to our initial question. Theorem D. 91' be a family of nonempty subsets of X. 91' is contained in some En' Proof' (i) = (ii). Fix Xo E X and for each En take B:;(x o)' (ii) = (i).