Supplementary Exercise 5.9 of IPS7e
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Weight gains for chicks in grams after one out of 20 diets:
  Chicks fed normal corn:      X ~ N(360,55)
  Chicks fed high-lysine corn: Y ~ N(385,50)
In both groups 20 chicks are used, and their weight gains are averaged. 
All weight gains are assumed independent.

(a)
Denote by Xmean and Ymean the mean (average) of the 20 chicks in the respective groups. 
To begin with we compute the means and variances for Xmean and Ymean (because the 
distribution of an average is different than the distribution of each individual value).

E(Xmean) = E(X) = 360; E(Ymean) = E(Y) = 385.
Var(Xmean) = 55^2/20 = 151.25; Var(Ymean) = 50^2/20 = 125.

Now we can compute the mean and standard deviation of Xmean-Ymean:

E(Ymean-Xmean) = E(Ymean) - E(Xmean) = 385-360 = 25
Var(Ymean-Xmean) = Var(Ymean) + Var(Xmean) = 151.25 + 125 = 276.25
sd(Ymean-Xmean) = sqrt(276.25) = 16.62

(b)
Most of this we already did in (a); the only new thing is that the variables
are normally distributed:

Xmean ~ N(360,55/sqrt(20)) = N(360,12.298)
Ymean ~ N(385,50/sqrt(20)) = N(385,11.180)
Ymean-Xmean ~ N(25,16.62), as computed above

(c)
P(Ymean-Xmean>=25) = 0.5 
(because Ymean-Xmean is normally distributed with mean 25, and the 
normal distribution is symmetrical around its mean).