A computer plots intervals that are centered at the sample percentage, and extend right and left from the sample percentage by the estimated standard error of the sample percentage. Recall that the SE of the sample percentage of n draws with replacement from a box that contains just zeros and ones is (p×(1−p))½/n½, where p is the percentage of ones in the box. For a box containing the numbers (1,1,0) with a sample size of 5, the SE(sample percentage) = (2/3×1/3)½/5½ = 0.21. In most situations in which we would need Statistics, we would not know the contents of the box, so we could not compute the true SE of the sample percentage. Instead, we would need to estimate the SD of the box—and thereby the SE of the sample percentage—from the sample. In fact, if we knew the SD of the box, there would only be two possible values of the population percentage.
If the SD of the box were 0.4899, the two possible values of the population would be (smaller of the two) _______? and (larger of the two) ____________?