Solution file for additional exercise 13.2
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Planning of a study on energy metabolism in cows, with the aim of 
comparing treatment and control groups. The outcome is BHB
concentrations in milk, and the calculation is based on a single
observation of each cow. The standard deviation is estimated at 650, and the
effect of interest is 300. The output from Minitab's power menu (see appendices 
for Stata and R listings) gives the result.

MTB > Power;
SUBC>   TTwo;
SUBC>     Difference 300;
SUBC>     Power .8;
SUBC>     Sigma 650.
Power and Sample Size 

2-Sample t Test

Testing mean 1 = mean 2 (versus not =)
Calculating power for mean 1 = mean 2 + difference
Alpha = 0.05  Assumed standard deviation = 650

            Sample  Target
Difference    Size   Power  Actual Power
       300      75     0.8      0.801778

The sample size is for each group.

Comments:
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To obtain a power of 0.8, it is required to have 75 cows in each group,
and therefore a total of 150 cows. The researchers chose a total of 200
cows, possibly because of anticipated dropouts; in a partly observational 
study like this one it is difficult to keep dropout rates low.

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Appendix: output from power calculation using Stata 13/14 menu:

. power twomeans 0, diff(300) sd(650)

Performing iteration ...

Estimated sample sizes for a two-sample means test
t test assuming sd1 = sd2 = sd
Ho: m2 = m1  versus  Ha: m2 != m1

Study parameters:

        alpha =    0.0500
        power =    0.8000
        delta =  300.0000
           m1 =    0.0000
           m2 =  300.0000
         diff =  300.0000
           sd =  650.0000

Estimated sample sizes:

            N =       150
  N per group =        75

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Appendix: output from power calculation using R function:

> power.t.test(delta=300, sd=650, sig.level=0.05, power=0.8, type="two.sample", alternative="two.sided")

     Two-sample t test power calculation 

              n = 74.66487
          delta = 300
             sd = 650
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: n is number in *each* group