A kind of plant has only blue flowers and white flowers. According to a genetic theory, the offspring
of a certain cross have a 75% chance to be blue-flowering and a 25% chance to be white-flowering,
independently of one another. Two hundred seeds of such a cross are raised, and 142 turn out to be
blue-flowering. Are the data consistent with the theory? Answer yes or no and explain.
X = 1 <- blue
X = 0 <- white
X ~ Bernoulli(0.75)
E(X) = p = 0.75
SD(X) = sqrt(p(1-p)) = sqrt(0.75*0.25) = 0.43
S = sum(X_k)
S ~ Bin(0.75, 200)
E(S) = Np = 200*0.75 = 150
SD(S) = sqrt(Np(1-p)) = sqrt(150*0.25) = sqrt(37.5) = 6.12
R = S/N
E(R) = p = 0.75
SD(R) = SE(X) = 1/sqrt(N) * sqrt(p(1-p)) = SD(X)/sqrt(N) = 0.43/sqrt(200) = 0.030
z = (142 - 150)/200*SD(E) = 0.04/0.030 = 1.306
p = P[z >= 1.306] = 0.097