Let x1, x2, and x3 be independent binary random variables taking 1 with probability p and zero with probability 1-p. Define two estimators p hat 1 = x bar and p hat 2 = (xbar/2) + (1/4), where xbar = (x1 + x2 + x3)/3.
For what values of p is the MSE of p hat 2 smaller than that of p hat 1?