Supplementary exercises 7.127 and 7.129 of IPS7e
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Data: Birth weight (in g) of babies born by women who tested positive on 
cocaine use compared to women who did not test positive. Note: this must be 
an observational study, not an experiment. Assume 75 women in each group.

Model: Two-sample inference based on normal distributions N(mu1,sigma)
and N(mu2,sigma); note that we assume the standard deviation to be the 
same in the two groups (a convenient assumption for planning purposes, 
but if in reality they differed, it would be good to take more subjects
in the group with the highest standard deviation).

The standard deviation is considered unknown and to be estimated by the
data, and our guess for the (common) value is sigma=650.

Hypotheses of interest are:
  H0: mu1=mu2 (no difference between groups)
  Ha: mu1<>mu2 (different birth weights in the cocaine group; a one-sided 
alternative can be argued as well but there might be so much doubt about 
the results from the previous study that the researcher opts for a more 
cautious approach)


7.127:
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Power calculation assuming a true population mean difference of
mu1-mu2=350.

MTB > Power;
SUBC>   TTwo;
SUBC>     Sample 75;
SUBC>     Difference 350;
SUBC>     Sigma 650;
SUBC>   GPCurve.
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
Difference    Size     Power
       350      75  0.905905

The sample size is for each group.

MTB > Power;
SUBC>   TTwo;
SUBC>     Sample 75;
SUBC>     Difference 350;
SUBC>     Sigma 650;
SUBC>     Alternative 1;
SUBC>   GPCurve.
Power and Sample Size 

2-Sample t Test

Testing mean 1 = mean 2 (versus >)
Calculating power for mean 1 = mean 2 + difference
Alpha = 0.05  Assumed standard deviation = 650

            Sample
Difference    Size     Power
       350      75  0.949227

The sample size is for each group.

Notes:
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The computed powers are 0.91 and 0.95, against a two-sided and one-sided alternative,
respectively. A study of this size is pretty likely to find a significant result
if the true magnitude of the difference is as presumed.


7.129:
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Solution for n=75 only.

The 95% confidence interval for mu1-mu2 is given by (when using the
pooled standard deviation):

  Xmean1-Xmean2 +- tstar*s*sqrt(1/n1 + 1/n2)

Here n1=n2=75
     df = (n1-1)+(n2-1) = 148,
     tstar = t_0.975(148) ~= t_0.975(100) = 1.984,
     (Minitab gives the exact value as 1.976)
and the margin of error is
   1.984*650*sqrt(2/75) = 210.6 ~= 210

It is seen that an observed difference of 350 would make the CI very far
from including zero, and would therefore correspond to a strongly significant result.
This may seem surprising in view of the power in the range 0.9-0.95 obtained
with n=75. However, a power calculation involves the true population difference,
not the observed sample difference. The true population difference being 350 does
not all guarantee that the sample difference is also close to 350, therefore 
it is a *stronger* requirement to have a fairly high power at a certain population
difference than having a margin of error for the CI of the same size.

Another way of saying this is that controlling the margin of error leads 
to a statement about what an observed mean difference of a certain size, here 
210 g, between the two groups tells us (namely, it is just at the significance 
cut-off), whereas a power calculation gives us the probability of getting a 
significant result from the two samples when the true population difference is 350 g.


Minitab command (to compute tstar) and listing:

MTB > InvCDF .975;
SUBC>   T 148.
Inverse Cumulative Distribution Function 

Student's t distribution with 148 DF

P( X <= x )        x
      0.975  1.97612


Minitab command to compute hypothetical confidence interval (because we
don't know the actual means) from which we can read off the margin of
error:

MTB > TwoT 75 0 650 75 0 650;
SUBC>   Confidence 95.0;
SUBC>   Test 0.0;
SUBC>   Alternative 0;
SUBC>   Pooled.
Two-Sample T-Test and CI 
                           SE
Sample   N  Mean  StDev  Mean
1       75     0    650    75
2       75     0    650    75

Difference = mu (1) - mu (2)
Estimate for difference:  0
95% CI for difference:  (-210, 210)
T-Test of difference = 0 (vs not =): T-Value = 0.00  P-Value = 1.000  DF = 148
Both use Pooled StDev = 650.0000

Comments:
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It is seen that the margin of error equals 210.