Section 9.2: Large-Sample Confidence Interval for a Population Proportion

Confidence Interval (CI) – for a population characteristic is an interval estimate of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence, the value of the characteristic is captured between the lower and upper endpoints of the interval. Confidence Level – associated with a confidence interval estimate is the success rate of the method used to construct the interval

Property of the Sampling Distribution of the Statistic p:

When n is large, approximately 95% of all samples of size n will result in a value of p that is within 1.96 standard deviation:

The interval can be used as long as: 1. np ≥ 10 and n(1 - p) ≥ The sample size is less than 10% of the population size if sampling is without replacement 3. The sample can be regarded as a random sample from the population of interest

Example Remember the example about affirmative action. We used π as a point of estimate. We found that π = 537/1013 =.530 np = (1013)(.530) = 537 ≥ 10 n(1 – p) = (1013)(.46) = 476 ≥ 10

Because both are larger than 10, we can use the large- sample interval.

Large-Sample Confidence Interval for π The general formula for a confidence interval for a population proportion π when 1.p is the sample proportion from a random sample 2.The sample size n is large (np ≥ 10 and n(1 – p) ≥ 10) 3.The sample size is small relative to the population size if the sample is selected without replacement (i.e., n is at most 10% of the population size) is

Standard Error – of a statistic is the estimated standard deviation of the statistic If the sampling distribution of a statistic is (at least approximately) normal, the bound on the error of estimation B associated with a 95% confidence interval is (1.96)(standard error of the statistic).

The sample size required to estimate a population proportion π to within an amount B with 95% confidence is The value of π may be estimated using prior information. In the absence of any such information, using π =.5 in this formula gives a conservatively large value for the required sample size (this value of π gives a larger n than would any other value)

Example A book includes a chapter on doctor- assisted suicide, caused a great deal of controversy in the medical community. Suppose that a survey of physicians is to be designed to estimate this proportion to within.05 with 95% confidence. How many primary-care physicians should be included in a random sample?