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Z-tests and t tests

data points should be independent from each other

z-test is preferable when
*n* is greater than 30.

the distributions should
be normal if *n* is low, if however *n*>30 the distribution
of the data does not have to be normal

the variances of the samples should be the same (F-test)

all individuals must be selected at random from the population

all individuals must have equal chance of being selected

sample sizes should be as equal as possible but some differences are allowed

data sets should be independent from each other except in the case of the paired-sample t-test

where *n*<30 the
t-tests should be used

the distributions should be normal for the equal and unequal variance t-test (K-S test or Shapiro-Wilke)

the variances of the samples should be the same (F-test) for the equal variance t-test

all individuals must be selected at random from the population

all individuals must have equal chance of being selected

sample sizes should be as equal as possible but some differences are allowed

if you do not find a significant difference in your data, you cannot say that the samples are the same

Z-test and t-test are basically the same; they compare between two means to suggest whether both samples come from the same population. There are however variations on the theme for the t-test. If you have a sample and wish to compare it with a known mean (e.g. national average) the single sample t-test is available. If both of your samples are not independent of each other and have some factor in common, i.e. geographical location or before/after treatment, the paired sample t-test can be applied. There are also two variations on the two sample t-test, the first uses samples that do not have equal variances and the second uses samples whose variances are equal.

It is well publicised that female students are currently doing better then male students! It could be speculated that this is due to brain size differences? To assess differences between a set of male students' brains and female students' brains a z or t-test could be used. This is an important issue (as I'm sure you'll realise lads) and we should use substantial numbers of measurements. Several universities and colleges are visited and a set of male brain volumes and a set of female brain volumes are gathered (I leave it to your imagination how the brain sizes are obtained!).

Excel can apply the z or t-tests to data arranged in rows or in columns, but the statistical packages nearly always use columns and are required side by side.

Degrees of freedom:

For the z-test degrees of freedom are not required since z-scores of 1.96 and 2.58 are used for 5% and 1% respectively.

For unequal and equal variance
t-tests = (*n*_{1} + *n*_{2}) - 2

For paired sample t-test = number of pairs - 1

The output from the z and t-tests are always similar and there are several values you need to look for:

You can check that the program
has used the right data by making sure that the means (1.81 and
1.66 for the t-test), number of observations (32, 32) and degrees
of freedom (62) are correct. The information you then need to
use in order to reject or accept your H_{O}, are the bottom
five values. The *t Stat* value is the calculated value relating
to your data. This must be compared with the two *t Critical*
values depending on whether you have decided on a one
or two-tail test (do not confuse these terms with the one
or two-way ANOVA). If the calculated value exceeds the critical
values the H_{O} must be rejected at the level of confidence
you selected before the test was executed. Both the one and two-tailed
results confirm that the H_{O} must be rejected and the
H_{A} accepted.

We can also use the P(T<=t)
values to ascertain the precise probability rather than the one
specified beforehand. For the results of the t-test above the
probability of the differences occurring by chance for the one-tail
test are 2.3x10^{-9} (from 2.3E-11 x 100). All the above
P-values denote very high significant differences.

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Ted Gaten Department of Biology gat@le.ac.uk Entry approved by the Head of Department. Last Updated: May 2000