we expect to see? We should see a similar normal distribution as in
A and B, but with a range intermediate between A and B. So, if
beak depth had no bearing on survival, our two samples should
have very similar means, just as any two random samples drawn
from the same population should have.
It is possible to get two random samples that have a statistically
significant difference just by chance, but it is unlikely. Figure 5
shows two groups randomly selected from a large population that,
by chance, have a statistically significant difference between their
means as determined by a t-test. If these two samples were selected
for an experiment, one as a control and the other as a treatment
group, and the independent variable actually had no effect, a t-test
would incorrectly result in the rejection of the null hypothesis. Statisticians call this a Type I error.
From the data provided in HHMI BioInteractive (2017), the
nonsurviving finches have a mean beak depth of 9.11 mm, with a
standard deviation of 0.88 mm. The surviving finches have a mean
beak depth of 9.67 mm, with a standard deviation of 0.84 mm. Is
this difference between the groups large enough to be considered
statistically significant, or are they just random samples from the
same population that, due to sampling error, have a difference in
their means? To answer this question, we perform a t-test on our
data. Inferential statistical procedures like the t-test have five basic
steps. The steps in the t-test are applied to the finch data as follows.
Steps 1 & 2: Choose the Appropriate Statistical Test &
State the Hypotheses
The t-test is appropriate for comparing the mean beak depths of
two small samples of continuous data points. The null hypothesis
states that any differences between the two groups will be small
and attributable mainly to chance factors, in this case primarily
genetic variation and sampling error. The alternative hypothesis
states that some factor caused a difference between the two groups;
in this case, differences in the mean beak depths of the survivors
and nonsurvivors influenced their survival. The formal statements
of the hypotheses are as follows.
• Null hypothesis (H0): There is no significant difference in mean
beak depth between the nonsurviving finches and finches that
survived the drought.
• Alternative hypothesis (H1): There is a significant difference in
mean beak depth between the nonsurviving finches and finches
that survived the drought.
Step 3: Choose the Decision Criterion
Next we determine what criterion to use in deciding whether the
difference between means is large enough to be statistically significant. As Fisher wrote, we must “forecast all possible results of the
Figure 4. Means of random samples drawn from a large population. Image captured from HHMI BioInteractive (2017; copyright
2015 Howard Hughes Medical Institute, used with permission; https://www.BioInteractive.org).