In the multinomial context, one would not expect to get 166 each of red and green M& M's and 167 each of brown, tan, orange, and yellow if one randomly chose 1000 from a vat with equal frequencies of the six colors. The value of X^2 for this selection would be .008, which is very close to zero, reflecting that the observed values are very close to the expected values. The probability of such a small X^2 if the selection was truly random is very small.

Being "too close" to the expected values can be quantified with the area to the left of the observed X^2. Subscripts such as .99 in the chi-squared table indicate that the area to the left is (in this case) .01. We identify a result as "to close" to the expected value if the observed X^2 lies within the left hand tail with the area of the specified level of significance.

*Example* If one found 19 B, 16 T, 17 G, 17 O, 16 Y, and 15 R candies in a bag which was allegedly randomly filled from a source with equal frequencies of all colors, The expected values would be 16.67 of each color, since there are 100 candies total. X^2 is readily calculated, and is equal to .560. There are five degrees of freedom. .554 .LT. .560 .LT. .831 (look at the 5 df row in the chi-squared table), hence the observed values are "too close" to the expected values at the .025 (= 1-.975) significance level, but not at the .01 (=1-.99) significance level. "Too close" in this case would indicate that the candies were selected to balance the colors,rather than at random.

Problem:

3) If a business man employs 12 Blacks, 7 Hispanics, and 31 Caucasians in a community which is 20% Black, 15% Hispanic, and 65% Caucasian; would you accuse him of using quotas? At what significance level?