two regions: (1) the region under the center of the t-distribution,
representing a small difference between the sample means that is
compatible with the null hypothesis; and (2) the region under the
extreme tails of the t-distribution, representing large differences
between the sample means that are very unlikely to occur if the null
hypothesis is true.
If we repeatedly take two random samples from the same
population that should have no differences between them and
compare their means with a t-test, 95% of the time the t-value
we calculate will fall in the 95% area of the distribution. Five
percent of the time, we will get a t-value that falls to the left or
right of the critical value indicating statistical significance (as in
Figure 5), even though both samples were chosen randomly
from the same population and there is no actual difference
Step 4: Calculate the t-Statistic
The t-statistic can be thought of as a ratio of “signal to noise.” The
expression in the numerator, (x̄ 1 − x̄ 2), the difference between the
two means, is the signal. The greater the difference, the stronger
the signal, the larger the t-value will be, and the more likely we will
achieve statistical significance.
But statistical significance also depends on the noise: the variability in the two sample datasets (Figure 8). The smaller the variability in the sample data, the more likely we are to find statistical
significance. The sample variances, the squares of the standard
deviations (s2 1 and s2 2), represent the variability in the beak depth
data: s2 1 ¼ ð0:84 mmÞ2 ¼ 0:71 mm2 and s2 2 ¼ ð0:88 mmÞ2 ¼
0:77mm2. Smaller variances make a smaller denominator, making
the t-value larger and making it more likely that we will achieve
Table 1. Partial table of critical t-values for
α = 0.05 (two-tailed t-test).
Figure 6. The t-distribution, which assumes that the null is
true.Mo dified from Statistics By Jim (https://statisticsbyjim.
Figure 7. Equation for calculating a t-value.
Figure 8. Examples of sample distributions with differing
degrees of variability (from Web Center for Social Research