NOTE: This page was developed using G*Power version 3.0.10. You candownload the current version of G*Power from https://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower.html. Youcan 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 particulartype of light bulb will last 850 hours on average with standard deviation of 50. A consumer protection group thinks that the manufacturer has overestimated thelifespan of their light bulbs by about 40 hours. How many light bulbs does theconsumer protection group have to test in order to make their point withreasonable 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 positivecorrelation between height and intelligence. If this is true, then the averageheight of a male graduate students on campus should be greater than theaverage height of American male adults in general. To test thistheory, one would randomly sample a small group of male graduate students.However, one would need to know how many male graduate students need tomeasured 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 theaverage lifespan of a light bulb. Here, the sample size (the number oflight bulbs to be tested) is the unknown to be solved for. We will need toidentify this variable for a given significance level and power.
A good start would be to list our known values and assumptions. Thebulbs’ stated longevity is 850, with detractors claiming 810. In otherwords, our null hypothesis H0 = 850, and the alternative hypothesis Ha=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 testis to show a discrepancy from the null hypothesis and not specifically agreater or lesser value, it is a two-tailed test.
Significance level sets the probability of Type 1 error; the probability thatthe null hypothesis will be rejected when it is, in fact, true.Conversely, power measures the probability that a Type 2 error will notoccur, a Type 2 error being the incidence of a false null hypothesis failing tobe rejected. In other words, power is the likelihood of the testappropriately 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, morespecifically, a means test involving a sample’s difference from a constant (howmuch do the reality of the bulbs differ from the manufacturer’s claim of 850hours?).
The type of power analysis being performed is noted to be an ‘APriori’ 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 inputstill requested is the effect size, or the difference of the null andhypothetical means divided by the standard deviation.
By clicking on the ‘Determine’ button to the left ofthe Effect size input, a new set of input cells is called up, for the nullhypothesis mean (here represented as MeanH0), the alternative mean (Mean H1), and the standard deviation (SD σ).As these numbers are known to us (850, 810, and 50), simply type themin and click ‘Calculate and transfer to main window’. As a result, theeffect level’s value (given as .8) is handily computed and inputted.
From there, a press of the ‘Calculate’ button in the main window produces thedesired sample size, among other statistics. These are, in descendingorder, the Noncentrality parameter δ, the Criticalt (the number of standard deviations from the null mean where an observationbecomes statistically significant), the number of degrees freedom, and thetest’s actual power. In addition, a graphical representation of thetest is shown, with the sampling distribution a dotted blue line, the populationdistribution represented by a solid red line, a red shaded area delineating theprobability of a type 1 error, a blue area the type 2 error, and a pair of greenlines evocating the critical points t.
To at last answer our question, the samplesize is shown to be 19. Thus, no fewer than nineteen light bulbs must betested in order to generate a statistically significant result (suggesting arejection of the null hypothesis, the manufacturer’s claim) with a power of .9.
To twist the initial question around, supposing only 10 light bulbs wereavailable for testing, what power would the test have, all else held constant?
This can be determined simply. The frame of the question is altered bysetting the type of power analysis from the ‘A Priori’ search for sample size toa ‘Post hoc’ pursuit of achieved power. Immediately, the input parametersreadjust to replace the power input with one for sample size. As all othervariables 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 thedegrees 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 notethat the test takes effect size into account, rather than the meansthemselves. As such, a null mean of 850 and an alternative mean of 810 areconsidered identical to a null mean of 810 and an alternative mean of 850, andare represented the same graphically. Thus, the graph displayed for ourexample is in fact a mirror image of what it should actually be, the nulldistribution being incorrectly to the left of the sampling distribution.It remains important to consider the numbers themselves and not be undulymisled.
As seen in the second half of the analysis, by adjusting the type of poweranalysis according to the values given and the values unknown, the requestedoutput can be generated for an unknown effect size, significance level, andimplied significance level with power, as well as the demonstrated ability toperform power and sample size calculations. In all cases, the unknown variableshould properly designated, followed by entering the givens in the inputparameters.
For more information on power analysis, please visit ourIntroduction to Power Analysisseminar.
