Thus, if y; is 10 percentage points above the mean, the expectation is that a.; = 0.4·10 = 4. But the conditional distribution of a.; has a standard deviation of 3.71-although the
expectation is of a 4 point industry effect, this estimate is only 1.08 standard deviations above zero. Using the normal distribution, FN(-1.08)= 0.14. That is, although industry i has an average return that is 10 percentage points above the mean, there remains a 0.14 probabillity that a.; is actually negative.
To reduce the chance of this type of error to 1.5 or less, the conditional estimate of a.; must be at least 1.64 standard deviations above zero (FN(l.64) = 0.95). That is, if one is to conclude a.;>0 with 95 percent confidence, the inequality