STA220- 5.4 Guided Notes
Large-Sample Confidence Interval for a Population Proportion
The point estimate for p, the population proportion of successes, is given by the proportion of
successes in a sample and is denoted by:
____________
where x is the number of successes in the sample and n is the number in the sample. The point
estimate for the number of failures is 𝑞̂ = 1 − 𝑝̂ .The symbols 𝑞̂ and 𝑝̂ are read as “p hat” and
“q hat.
Example 1:
In July of 2008, a Quinnipiac University Poll asked 1783 registered voters nationwide whether
they favored or opposed the death penalty for persons convicted of murder. 1123 were in
favor. Obtain a 90% confidence interval for the proportion of registered voters nationwide who
are in favor of the death penalty for persons convicted of murder.
Example 2:
The graph shown below is from a survey of 935 adults. Construct a 99% confidence interval for
the proportion of adults who think that airplanes are the safest mode of transportation.
CAUTION!!!!!!!
Unless ‘n’ is extremely large, the large-sample procedure presented in this section performs
poorly when ‘p’ is near 0 or near 1.
If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results where
Example 3:
Suppose in a particular year the percentage of firms declaring bankruptcy that had shown
profits the previous year is .002. If 100 firms are sampled and one had declared bankruptcy,
what is the 95% CI on the proportion of profitable firms that will tank the next year?
Example 4:
According to True Odds: How Risk Affects Your Everyday Life (Walsh, 1997), the probability of
being the victim of a violent crime is less than 0.01. Suppose that, in a random sample of 200
Americans, 3 were victims of a violent crime. Use a 95% confidence interval to estimate the
true proportion of Americans who were victims of a violent crime.