NOTE:  This page was developed using G*Power version 3.0.10.  You can
download the current version of G*Power from https://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower.html
.  You
can also find help files, the manual and the user guide on this website.

Examples

Example 1.  A company that manufactures light bulbs claims that a particular
type of light bulb will last 850 hours on average with standard deviation of 50.  A consumer protection group thinks that the manufacturer has overestimated the
lifespan of their light bulbs by about 40 hours.  How many light bulbs does the
consumer protection group have to test in order to make their point with
reasonable confidence?

Example 2.  It has been estimated that the average height of American male adults is 70 inches. I t has also been postulated that there is a positive
correlation between height and intelligence.  If this is true, then the average
height of a male graduate students on campus should be greater than the
average height of American male adults in general.  To test this
theory, one would randomly sample a small group of male graduate students.
However, one would need to know how many male graduate students need to
measured such that the hypothesis can be reasonable tested.

Prelude to the power analysis

For the power analysis below, we are going to focus on Example 1, testing the
average lifespan of a light bulb.  Here, the sample size (the number of
light bulbs to be tested) is the unknown to be solved for.  We will need to
identify this variable for a given significance level and power.

A good start would be to list our known values and assumptions.  The
bulbs’ stated longevity is 850, with detractors claiming 810.  In other
words, our null hypothesis H₀ = 850, and the alternative hypothesis H_a=
810.  It is also of great importance to note that the standard deviation is 50, as not all light bulbs are created equal.  Additionally, as the test
is to show a discrepancy from the null hypothesis and not specifically a
greater or lesser value, it is a two-tailed test.

Significance level sets the probability of Type 1 error; the probability that
the null hypothesis will be rejected when it is, in fact, true.
Conversely, power measures the probability that a Type 2 error will not
occur, a Type 2 error being the incidence of a false null hypothesis failing to
be rejected.  In other words, power is the likelihood of the test
appropriately rejecting H₀.  For this example,
we will choose a significance level of .05 and a power of .9.

Power analysis

Immediately, we can put our known measures into G*Power’s interface.

We begin by indicating that we are performing a t-test, and, more
specifically, a means test involving a sample’s difference from a constant (how
much do the reality of the bulbs differ from the manufacturer’s claim of 850
hours?).

The type of power analysis being performed is noted to be an ‘A
Priori’ analysis, a determination of sample size.  From there, we can input the number of tails,
the value of our chosen significance level (α), and the power; 2, .05, and .9,
respectively.  The only input
still requested is the effect size, or the difference of the null and
hypothetical means divided by the standard deviation.

By clicking on the ‘Determine’ button to the left of
the Effect size input, a new set of input cells is called up, for the null
hypothesis mean (here represented as Mean
H0), the alternative mean (Mean H1), and the standard deviation (SD σ).
As these numbers are known to us (850, 810, and 50), simply type them
in and click ‘Calculate and transfer to main window’.  As a result, the
effect level’s value (given as .8) is handily computed and inputted.

From there, a press of the ‘Calculate’ button in the main window produces the
desired sample size, among other statistics.  These are, in descending
order, the Noncentrality parameter δ, the Critical
t (the number of standard deviations from the null mean where an observation
becomes statistically significant), the number of degrees freedom, and the
test’s actual power.  In addition, a graphical representation of the
test is shown, with the sampling distribution a dotted blue line, the population
distribution represented by a solid red line, a red shaded area delineating the
probability of a type 1 error, a blue area the type 2 error, and a pair of green
lines evocating the critical points t.

To at last answer our question, the sample
size is shown to be 19.  Thus, no fewer than nineteen light bulbs must be
tested in order to generate a statistically significant result (suggesting a
rejection of the null hypothesis, the manufacturer’s claim) with a power of .9.

To twist the initial question around, supposing only 10 light bulbs were
available for testing, what power would the test have, all else held constant?

This can be determined simply.  The frame of the question is altered by
setting the type of power analysis from the ‘A Priori’ search for sample size to
a ‘Post hoc’ pursuit of achieved power.  Immediately, the input parameters
readjust to replace the power input with one for sample size.  As all other
variables remain as previous, the new measure of sample size, 10, is entered in.

Making use of the Calculate button, we receive the new output parameters.

These include the Noncentrality parameter δ, the Critical t, and the
degrees freedom as before, in addition to Power, here measuring 0.616233,
having decreased from .9 due to the smaller sample.

Discussion

In reference to the initial question and its outcome, it is important to note
that the test takes effect size into account, rather than the means
themselves.  As such, a null mean of 850 and an alternative mean of 810 are
considered identical to a null mean of 810 and an alternative mean of 850, and
are represented the same graphically.  Thus, the graph displayed for our
example is in fact a mirror image of what it should actually be, the null
distribution being incorrectly to the left of the sampling distribution.
It remains important to consider the numbers themselves and not be unduly
misled.

As seen in the second half of the analysis, by adjusting the type of power
analysis according to the values given and the values unknown, the requested
output can be generated for an unknown effect size, significance level, and
implied significance level with power, as well as the demonstrated ability to
perform power and sample size calculations. In all cases, the unknown variable
should properly designated, followed by entering the givens in the input
parameters.

For more information on power analysis, please visit our
Introduction to Power Analysis
seminar.