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Chapter 12: Leaving Room for a Margin of Error
the statistic used to report the results is the proportion of people from the
sample who fell into a certain group (for example, the “approve” group).
This is known as the sample proportion. You find this number by taking the
number of people in the sample that fell into the group of interest, divided by
the sample size, n.
Along with the sample proportion, you need to report a margin of error. The
general formula for margin of error for the sample proportion (if certain
conditions are met) is
the sample size, and z* is the appropriate z*-value for your desired level of
confidence (from Table 12-1). Here are the steps for calculating the margin of
error for a sample proportion:
1. Find the sample size, n, and the sample proportion, .
The sample proportion is the number in the sample with the character-
istic of interest, divided by n. , where is the sample proportion, n is 185
2. Multiply the sample proportion by .
3. Divide the result by n.
4. Take the square root of the calculated value.
You now have the standard error, .
5. Multiply the result by the appropriate z*-value for the confidence
level desired.
Refer to Table 12-1 for the appropriate z*-value. If the confidence level is
95%, the z*-value is 1.96.
Looking at the example involving whether Americans approve of the presi-
dent, you can find the actual margin of error. First, assume you want a 95%
level of confidence, so z* = 1.96. The number of Americans in the sample who
said they approve of the president was found to be 520. This means that the
sample proportion, , is 520 ÷ 1,000 = 0.52. (The sample size, n, was 1,000.)
The margin of error for this polling question is calculated in the following way:
According to this data, you conclude with 95% confidence that 52% of all
Americans approve of the president, plus or minus 3.1%.
Two conditions need to be met in order to use a z*-value in the formula for
margin of error for a sample proportion:
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