SECTION 6.3 Confidence Intervals for Population Proportions 321 Confidence Intervals for a Population Proportion Constructing a confidence interval for a population proportion p is similar to constructing a confidence interval for a population mean. You start with a point estimate and calculate a margin of error. A c@confidence interval for a population proportion p is np - E 6 p 6 np + E where E = zcBnpnq n . Margin of error for p The probability that the confidence interval contains p is c, assuming that the estimation process is repeated a large number of times. DEFINITION In Section 5.5, you learned that a binomial distribution can be approximated by a normal distribution when np Ú 5 and nq Ú 5. When nnp Ú 5 and nnq Ú 5, the sampling distribution of np is approximately normal with a mean of mnp = p Mean of the sample proportions and a standard error of snp = Apq n . Standard error of the sample proportions aNotice snp = s n = 1npq n = 1npq 2n2 = Anpq n2 = Apq n .b Constructing a Confidence Interval for a Population Proportion In Words In Symbols 1. Identify the sample statistics n and x. 2. Find the point estimate np. np = x n 3. Verify that the sampling distribution nnp Ú 5, nnq Ú 5 of np can be approximated by a normal distribution. 4. Find the critical value zc that Use Table 4 in Appendix B. corresponds to the given level of confidence c. 5. Find the margin of error E. E = zcBnpnq n 6. Find the left and right endpoints Left endpoint: np - E and form the confidence interval. Right endpoint: np + E Interval: np - E 6 p 6 np + E GUIDELINES In Step 4 above, note that the critical value zc is found the same way it was found in Section 6.1, by either using Table 4 in Appendix B or using technology. Picturing the World A poll surveyed 2140 U.S. adults about the privacy of their personal information. Of those surveyed, 813 said that they think it is possible to go about daily life today without having companies collect data about them. (Adapted from Pew Research Center) Do You Think It Is Possible to Go About Daily Life Today Without Having Companies Collect Data About You? Yes 813 No 1327 Find a 90% confidence interval for the population proportion of people who think it is possible to go about daily life today without having companies collect data about them. Tech Tip Here are instructions for constructing a confidence interval for a population proportion on a TI-84 Plus. STAT Choose the TESTS menu. A: 1–PropZInt . . . Enter the values of x, n, and the level of confidence c (C-Level). Then select Calculate.
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