Final answer:
A confidence interval for a population proportion is calculated using the sample proportion, sample size, and associated z-score. The interval is estimated by adding and subtracting the margin of error from the sample proportion, with both np' and nq' being greater than five for accuracy.
Step-by-step explanation:
Calculating Confidence Interval for Population Proportion
To calculate a confidence interval for a population proportion, you first need the sample proportion (p') and the sample size (n). From these, you can calculate the estimated proportion of failures (q' = 1 - p'). The standard error (SE) of the proportion is then calculated using the formula SE = √[(p')(q') / n]. For a 95% confidence interval, you also need the z-score associated with this confidence level, which is typically 1.96 for a two-tailed interval.
Therefore, the 95% confidence interval can be calculated as p' ± Z * SE, where Z is the z-score. To ensure accuracy, the rule of thumb is that both np' and nq' should be greater than five.
For the example given, with a sample of 500 city residents where 280 own an automobile, the sample proportion (p') is 280/500 = 0.56, and q' is 1 - 0.56 = 0.44. The SE would be √[(0.56)(0.44) / 500]. Multiplying this SE by the z-score of 1.96 provides the margin of error, which is then added to and subtracted from p' to obtain the confidence interval.
Once the upper and lower bounds of the interval are determined, they can be expressed in the form: (lower bound, upper bound). This interval estimates the true population proportion with a specified level of confidence.
In terms of sample size, if you want the margin of error to be within a certain range, you can use the formula for the error bound of a population proportion (EBP) to work backward and find the minimum sample size required.