Supplementary Exercise 6.115 of IPS7e
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Data: SAT scores of 500 high school students selected as a simple random
sample from a certain population. Interest is in whether the population
mean is larger than 450. 

Model: A simple random sample from a population with mean and standard
deviation sigma, where sigma is known (sigma=100). In this situation it
might actually be reasonable to assume the standard deviation to be
known because large groups of students have been tested; the assumption
is then that this particular population has the same standard deviation
as the more general population.

Hypotheses of interest are:
  H0: mu=450 
  Ha: mu>450 (because interest is whether this population performs better
              than 450; this motivation is not entirely clear to me)

The test will be carried out at a 1% significance level (somewhat
unusually). A power calculation for a hypothesized true mean of mu=460, 
corresponding to a hypothesized true difference between true mean and 
tested mean of 460-450=10.

Minitab computation of power:

MTB > Power;
SUBC>   ZOne;
SUBC>     Sample 500;
SUBC>     Difference 10;
SUBC>     Sigma 100;
SUBC>     Alternative 1;
SUBC>     Alpha 0.01.
Power and Sample Size 

1-Sample Z Test

Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.01  Assumed standard deviation = 100

            Sample
Difference    Size     Power
        10     500  0.464032

Comments:
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The computed power is 0.46. This means that there is a 46% chance of
rejecting the null hypothesis (mu=450) based on a sample of 500 students
if the true mean was 460. With such a fairly low power, we need a bit of
"luck" to get significance, and a non-significant result could easily
happen by chance (even if the null hypothesis is false). It would be
fair to say that the sample of 500 students has insufficient power.