Download Inferences About the Difference Between Two Independent Sa

Inferences About the Difference Between Two Independent
Samples: Unpaired
'Student's' t Test is one of the most commonly used techniques for testing a hypothesis
on the basis of a difference between sample means. Explained in layman's terms, the t
test determines a probability that two populations are the same with respect to the
variable tested.
For example, suppose you collected data on the heights of male basketball and football
players, and compared the sample means using the t test. A probability of 0.4 would mean
that there is a 40% likelihood that you cannot distinguish a group of basketball players
from a group of football players by height alone. That's about as far as the t test or any
statistical test, for that matter, can take you. If you calculate a probability of 0.05 or
less, then you can reject the null hypothesis (that is, you can conclude that the two groups
of athletes can be distinguished by height).
To the extent that there is a small probability that you are wrong, you haven't proven a
difference, though. There are differences among popular, mathematical, philosophical,
legal, and scientific definitions of proof. Never use the word “prove” or “proof” – please
use the term “support”. Don't make the error of reporting your results as proof (or
disproof) of a hypothesis. No experiment is perfect, and proof in the strictest sense
requires perfection. We are not capable of being perfect... or I’m not.
Make sure you understand the concepts of experimental error and single variable
statistics before you go through this part. Leaves were collected from wax-leaf ligustrum
grown in shade and in full sun. The thickness in micrometers of the palisade layer was
recorded for each type of leaf. Thicknesses of 7 sun leaves were reported as: 150, 100,
210, 300, 200, 210, and 300, respectively. Thicknesses of 7 shade leaves were reported as
120, 125, 160, 130, 200, 170, and 200, respectively. The mean ± standard deviation for sun
leaves was 210 ± 73 micrometers and for shade leaves it was158 ± 34 micrometers. Note
that since all data were rounded to the nearest micrometer, it is inappropriate to include
decimal places in either the mean or standard deviation.
For the t test for independent samples you do not have to have the same number of data
points in each group. We have to assume that the population follows a normal distribution
(small samples have more scatter and follow what is called a t distribution). Corrections
can be made for groups that do not show a normal distribution (skewed samples, for
example - note that the word 'skew' has a specific statistical meaning, so don't use it as a
synonym for 'messed up').
The t test can be performed knowing just the means, standard deviation, and number of
data points. Note that the raw data must be used for the t test or any statistical test, for
that matter. If you record only means in your notebook, you lose a great deal of
information and usually render your work invalid. The two sample t test yields a statistic t,
in which
X-bar, of course, is the sample mean, and s is the sample standard deviation. Note that the
numerator of the formula is the difference between means. The denominator is a
measurement of experimental error in the two groups combined. The wider the difference
between means, the more confident you are in the data. The more experimental error you
have, the less confident you are in the data. Thus the higher the value of t, the greater
the confidence that there is a difference.
To understand how a precise probability value can be attached to that confidence you need
to study the mathematics behind the t distribution in a formal statistics course. The value
t is just an intermediate statistic. Probability tables have been prepared based on the t
distribution originally worked out by W.S. Gossett (see below). To use the table provided,
find the critical value that corresponds to the number of degrees of freedom you have
(degrees of freedom = number of data points in the two groups combined, minus 2). If t
exceeds the tabled value, the means are significantly different at the probability level
that is listed. When using tables report the lowest probability value for which t exceeds
the critical value. Report as 'p < (probability value).'
In the example, the difference between means is 52, A = 14/49, and B = 3242.5. Then t =
1.71 (rounding up). There are (7 + 7 -2) = 12 degrees of freedom, so the critical value for p
= 0.05 is 2.18. 1.71 is less than 2.18, so we cannot reject the null hypothesis that the two
populations have the same palisade layer thickness. So now what? If the question is very
important to you, you might collect more data. With a well designed experiment, sufficient
data can overcome the uncertainty contributed by experimental error, and yield a
significant difference between samples, if one exists.
If you have lots of data and the probability value becomes smaller but still does not reach
the 'magic' number 0.05, should you keep collecting data until it does? At this point,
consider the biological significance of the question. If you did find a difference of 0.1%
between palisade layers of sun and shade leaves respectively, just how important could it
be?
When reporting results of a statistical analysis, always identify what data sets you
compared, what test was used, and for most quantitative data report mean, standard
deviation, and the probability values. Make sure the outcome of the analysis is clearly
reported. Some spreadsheet programs include the t test for independent variables as a
built-in option. Even without a built-in option, is so easy to set up a spreadsheet to do a
paired t test that it may not be worth the expense and effort to buy and learn a dedicated
statistics software program, unless more complicated statistics are needed.