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 H_{0} = 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 H_{0}. 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.