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Part IV: Guesstimating and Hypothesizing with Confidence
called a confidence interval for the population mean, . Its formula depends
on whether certain conditions are met. I split the conditions into two cases,
illustrated in the following sections.
Case 1: Population standard
deviation is known
In Case 1, the population standard deviation is known. The formula for a
confidence interval (CI) for a population mean in this case is ,
where is the sample mean, is the population standard deviation, n is the
sample size, and z* represents the appropriate z*-value from the standard
normal distribution for your desired confidence level. (Refer to Table 13-1 for
values of z* for the given confidence levels.)
In this case, the data either have to come from a normal distribution, or if not,
then n has to be large enough (at least 30 or so) for the Central Limit Theorem
to kick in (see Chapter 11), allowing you to use z*-values in the formula.

To calculate a CI for the population mean (average), under the conditions for
Case 1, do the following:

1. Determine the confidence level and find the appropriate z*-value.
Refer to Table 13-1.

2. Find the sample mean ( ) for the sample size (n).
Note: The population standard deviation is assumed to be a known
value, .
3. Multiply z* times and divide that by the square root of n.
This calculation gives you the margin of error.

4. Take plus or minus the margin of error to obtain the CI.
The lower end of the CI is minus the margin of error, whereas the
upper end of the CI is plus the margin of error.

For example, suppose you work for the Department of Natural Resources and
you want to estimate, with 95% confidence, the mean (average) length of wall-
eye fingerlings in a fish hatchery pond.
1. Because you want a 95% confidence interval, your z*-value is 1.96.

2. Suppose you take a random sample of 100 fingerlings and determine
that the average length is 7.5 inches; assume the population standard
deviation is 2.3 inches. This means , , and n = 100.

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