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Chapter 13: Confidence Intervals: Making Your Best Guesstimate

3. Multiply 1.96 times 2.3 divided by the square root of 100 (which is 10). The
margin of error is, therefore, ± 1.96 ∗ (2.3 ÷ 10) = 1.96 ∗ 0.23 = 0.45 inches.
4. Your 95% confidence interval for the mean length of walleye fingerlings

in this fish hatchery pond is 7.5 inches ± 0.45 inches. (The lower end of
the interval is 7.5 – 0.45 = 7.05 inches; the upper end is 7.5 + 0.45 = 7.95
inches.)
After you calculate a confidence interval, make sure you always interpret it in 203
words a non-statistician would understand. That is, talk about the results in
terms of what the person in the problem is trying to find out — statisticians
call this interpreting the results “in the context of the problem.” In this exam-
ple you can say: “With 95% confidence, the average length of walleye finger-
lings in this entire fish hatchery pond is between 7.05 and 7.95 inches, based
on my sample data.” (Always be sure to include appropriate units.)
Case 2: Population standard deviation
is unknown and/or n is small
In many situations, you don’t know , so you estimate it with the sample stan-
dard deviation, s; and/or the sample size is small (less than 30), and you can’t
be sure your data came from a normal distribution. (In the latter case, the
Central Limit Theorem can’t be used; see Chapter 11.) In either situation, you
can’t use a z*-value from the standard normal (Z-) distribution as your critical
value anymore; you have to use a larger critical value than that, because of
not knowing what is and/or having less data.

The formula for a confidence interval for one population mean in Case 2
is , where t* is the critical t*-value from the t-distribution
n – 1
with n – 1 degrees of freedom (where n is the sample size). The t*-values for
common confidence levels are found using the last row of the t-table (in the
appendix). Chapter 10 gives you the full details on the t-distribution and how
to use the t-table.

The t-distribution has a similar shape to the Z-distribution except it’s flatter
and more spread out. For small values of n and a specific confidence level,
the critical values on the t-distribution are larger than on the Z-distribution,
so when you use the critical values from the t-distribution, the margin of error
for your confidence interval will be wider. As the values of n get larger, the
t*-values are closer to z*-values. (Chapter 10 gives you the full details on the
t-distribution and its relationships to the Z-distribution.)

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