3.3 Constructing a Confidence Interval for a Population Proportion

Identify an appropriate confidence interval procedure including the parameter for a population proportion.

• A confidence interval is an interval estimate for a population parameter. Based on the sample proportion, a confidence interval can be calculated to estimate the value of a single population proportion. The appropriate confidence interval procedure is a one-sample z-interval for a population proportion.
• The parameter for a confidence interval for a population proportion should reference the proportion, the response variable, and the population in context.

Justify the appropriateness of constructing a confidence interval for a population proportion by verifying conditions.

• A one-sample z-interval for a population proportion requires that three conditions be met:
  • i. The randomization condition—the data should be collected using a random sample.
  • ii. The 10% condition—when sampling without replacement, the population size must be at least 10 times larger than the sample size nN10% , where N is the size of the population and n is the sample size.
  • iii. The normality condition—the observed number of successes, , and the observed number of failures, 1 , should be at least 10.

Calculate an appropriate confidence interval for a population proportion.

• z denotes a critical value, such that z and z represent the boundaries enclosing the middle C% of the standard normal distribution, in which C% is an approximate confidence level with which the population proportion is estimated.
• An interval estimate can be constructed as point estimate (margin of error). For a population proportion, the one-sam p ple z-interval estimate is p 1 z n .

Calculate the standard error and margin of error of a sample statistic for a confidence interval for a population proportion, and estimate a given sample size from the margin of error.

• The standard error (SE) of a statistic is an estimate of the standard deviation of the sampling distribution of the statistic. The standard error of the sample proportion p is SE pp np 1 .
• The standard error quantifies the typical amount that a statistic will vary from the value of the corresponding population parameter.
• The margin of error of p is half the width of the confidence interval and is calculated as the critical value z times the standard error (SE) of p, which equals p 1 z n . p
• The formula for the margin of error (MOE) can be rearranged to solve for n, n zp p MOE 2  1 2 , the minimum sample needed to achieve a given margin of error. For this purpose, if p is not defined or unable to be calculated, use p=0.5 in order to find the upper bound for the sample size that will result in a given mar gin of error. Inference for Categorical Data: Proportions UNIT 3 90