Russel-Research-Method-in-Anthropology

Sampling Theory 181
Look up what’s called the critical value of t in appendix C. In the column
for .025, we see that the value is 2.571 with 5 degrees of freedom. Degrees of
freedom are one less than the size of the sample, so for a sample of six we
need a t statistic of 2.571 to attain 95% confidence. (The concept of degrees
of freedom is described further in chapter 19 in the section on t-tests. And
keep in mind that we’re interested in the modulus, or absolute value of t. A
value of –2.571 is just as statistically significant as a value of 2.571.)
So, with small samples—which, for practical purposes, means less than 30
units of analysis—we use appendix C (for t) instead of appendix B (for z) to
determine the confidence limits around the mean of our estimate. You can see
from appendix C that for large samples—30 or more—the difference between
the t and the z statistics is negligible. (The t-value of 2.042 for 30 degrees of
freedom—which means a sample of 31—is very close to 1.96.)
The Catch
Suppose that instead of estimating the income of a population with a sample
of 100, we use a sample of 10 and get the same result—RM12,600 and a standard
deviation of RM4,000. For a sample this size, we use the t distribution.
With 9 degrees of freedom and an alpha value of .025, we have a t value of
2.262. For a normal curve, 95% of all scores fall within 1.96 standard errors
of the mean. The corresponding t value is 2.262 standard errors. Substituting
in the formula, we get:
RM12,600 2.262 RM4,000 RM9,739 to RM15,461
10
But there’s a catch. With a large sample (greater than 30), we know from the
central limit theorem that the sampling distribution will be normal even if the
population isn’t. Using the t distribution with a small sample, we can calculate
the confidence interval around the mean of our sample only under the assumption
that the population is normally distributed.
In fact, looking back at figure 7.5, we know that the distribution of real data
is not perfectly normal. It is somewhat skewed (more about skewed distributions
in chapter 19). In real research, we’d never take a sample from a set of
just 50 data points—we’d do all our calculations on the full set of the actual
data. When we take samples, it’s because we don’t know what the distribution
of the data looks like. And that’s why sample size counts.
Calculating Sample Size for Estimating Proportions
And now for proportions. What we’ve learned so far about estimating the
mean of continuous variables (like income and percentages) is applicable to
the estimation of proportions as well.