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Part IV: Guesstimating and Hypothesizing with Confidence
Case 2: Population standard deviations are
unknown and/or sample sizes are small
In many situations, you don’t know
the sample standard deviations, s , and s ; and/or the sample sizes are small
(less than 30) and you can’t be sure whether your data came from a normal
distribution. 1 2 , and you estimate them with
A confidence interval for the difference in two population means under
Case 2 is , where t* is the critical value
from the t-distribution with n + n – 2 degrees of freedom; n and n are the
1 2 1 2
two sample sizes, respectively; and s and s are the two sample standard
1 2
deviations. This t*-value is found on the t-table (in the appendix) by
intersecting the row for df = n + n – 2 with the column for the confidence
1 2
level you need, as indicated by looking at the last row of the table. (See
Chapter 10.) Here we assume the population standard deviations are similar;
if not, modify by using the standard error and degrees of freedom. See the
end of the section on comparing two means in Chapter 15.
In the corn example from Case 1, suppose the mean cob lengths of the two
brands of corn, Corn-e-stats (group 1) and Stats-o-sweet (group 2), are the same
as they were before: inches. But this time you don’t know
the population standard deviations, so you use the sample standard deviations
instead — suppose they turn out to be s = 0.40 and s = 0.50 inches, respectively.
1 2
Suppose the sample sizes, n and n , are each only 15 in this case.
1 2
Calculating the CI, you first need to find the t*-value on the t-distribution with
(15 + 15 – 2) = 28 degrees of freedom. (Assume the confidence level is still 95%.)
Using the t-table (in the appendix), look at the row for 28 degrees of freedom
and the column representing a confidence level of 95% (see the labels on the
last row of the table); intersect them and you see t* = 2.048. Using the rest of
28

the information you are given, the confidence interval for the difference in mean
cob length for the two brands is
.

That means a 95% CI for the difference in the mean cob lengths of these two
brands of corn in this situation is (0.0727, 1.9273) inches, with Corn-e-stats
coming out on top. (Note: This CI is wider than what was found in Case 1, as
expected.)

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