Learning Outcomes
- Using the formula for creating a confidence interval or technology, construct a confidence interval for a population proportion using the “plus-four” method
“Plus Four” Confidence Interval for p
There is a certain amount of error introduced into the process of calculating a confidence interval for a proportion. Because we do not know the true proportion for the population, we are forced to use point estimates to calculate the appropriate standard deviation of the sampling distribution. Studies have shown that the resulting estimation of the standard deviation can be flawed.
Fortunately, there is a simple adjustment that allows us to produce more accurate confidence intervals. We simply pretend that we have four additional observations. Two of these observations are successes and two are failures. The new sample size, then, is n + 4, and the new count of successes is x + 2.
Computer studies have demonstrated the effectiveness of this method. It should be used when the confidence level desired is at least 90% and the sample size is at least ten.
Example 3
A random sample of 25 statistics students was asked: “Have you smoked a cigarette in the past week?” Six students reported smoking within the past week. Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke.
The first solution is step-by-step.
The second solution uses a function of the TI-83, 83+, or 84 calculators.
try it 3
Out of a random sample of 65 freshmen at State University, 31 students have declared a major. Use the “plus-four” method to find a 96% confidence interval for the true proportion of freshmen at State University who have declared a major.
The first solution is step-by-step.
The second solution uses a function of the TI-83, 83+, or 84 calculators.
Example 4
The Berkman Center for Internet & Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. In a group of 50 teens, 13 reported having more than 500 friends on Facebook. Use the “plus-four” method to find a 90% confidence interval for the true proportion of teens who would report having more than 500 Facebook friends.
The first solution is step-by-step.
The second solution uses a function of the TI-83, 83+, or 84 calculators.
try it 4
The Berkman Center Study referenced in Example 4 talked to teens in smaller focus groups, but also interviewed additional teens over the phone. When the study was complete, 588 teens had answered the question about their Facebook friends with 159 saying that they have more than 500 friends. Use the “plus-four” method to find a 90% confidence interval for the true proportion of teens that would report having more than 500 Facebook friends based on this larger sample. Compare the results to those in Example 4.
The first solution is step-by-step.
The second solution uses a function of the TI-83, 83+, or 84 calculators.
Conclusion
The confidence interval for the larger sample is narrower than the interval from Example 4. Larger samples will always yield more precise confidence intervals than smaller samples. The “plus-four” method has a greater impact on the smaller sample. It shifts the point estimate from 0.26 (13/50) to 0.278 (15/54). It has a smaller impact on the EPB, changing it from 0.102 to 0.100. In the larger sample, the point estimate undergoes a smaller shift: from 0.270 (159/588) to 0.272 (161/592). It is easy to see that the plus-four method has the greatest impact on smaller samples.
