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Part IV: Guesstimating and Hypothesizing with Confidence

If you are wondering where this formula for sample size came from, it’s actu-
ally created with just a little math gymnastics. Take the margin of error for-
mula (which contains n), fill in the remaining variables in the formula with
numbers you glean from the problem, set it equal to the desired MOE, and
solve for n.
Determining the Confidence Interval
for One Population Proportion
When a characteristic being measured is categorical — for example, opinion
on an issue (support, oppose, or are neutral), gender, political party, or type
of behavior (do/don’t wear a seatbelt while driving) — most people want to
estimate the proportion (or percentage) of people in the population that fall
into a certain category of interest. For example, consider the percentage of
people in favor of a four-day work week, the percentage of Republicans who
voted in the last election, or the proportion of drivers who don’t wear seat
belts. In each of these cases, the object is to estimate a population propor-
tion, p, using a sample proportion, , plus or minus a margin of error. The
result is called a confidence interval for the population proportion, p.

The formula for a CI for a population proportion is , where is
the sample proportion, n is the sample size, and z* is the appropriate value
from the standard normal distribution for your desired confidence level.
Refer to Table 13-1 for values of z* for certain confidence levels.

To calculate a CI for the population proportion:

1. Determine the confidence level and find the appropriate z*-value.
Refer to Table 13-1 for z*-values.
2. Find the sample proportion, , by dividing the number of people in
the sample having the characteristic of interest by the sample size (n).
Note: This result should be a decimal value between 0 and 1.

3. Multiply and then divide that amount by n.
4. Take the square root of the result from Step 3.
5. Multiply your answer by z*.
This step gives you the margin of error.
6. Take plus or minus the margin of error to obtain the CI; the lower
end of the CI is minus the margin of error, and the upper end of the
CI is plus the margin of error.

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