Confidence Intervals for Population Proportions 6.3 320 CHAPTER 6 Confidence Intervals What You Should Learn How to find a point estimate for a population proportion How to construct and interpret confidence intervals for a population proportion How to determine the minimum sample size required when estimating a population proportion Point Estimate for a Population Proportion Confidence Intervals for a Population Proportion Finding a Minimum Sample Size Point Estimate for a Population Proportion Recall from Section 4.2 that the probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. In this section, you will learn how to estimate a population proportion p using a confidence interval. As with confidence intervals for m, you will start with a point estimate. The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by np = x n Sample proportion where x is the number of successes in the sample and n is the sample size. The point estimate for the population proportion of failures is nq = 1 - np. The symbols np and nq are read as “p hat” and “q hat.” DEFINITION Finding a Point Estimate for p In a recent survey of 540 U.S. adults, 378 said that they plan on traveling this summer. Find a point estimate for the population proportion of U.S. adults who plan on traveling this summer. (Adapted from TravelDailyNews International) SOLUTION The number of successes is the number of adults who plan on traveling this summer, so x = 378. The sample size is n = 540. So, the sample proportion is np = x n Formula for sample proportion = 378 540 Substitute 378 for x and 540 for n. = 70%. Divide and write as a percent. So, the point estimate for the population proportion of U.S. adults who plan on traveling this summer is 70% or 0.70. TRY IT YOURSELF 1 A poll surveyed 1050 U.S. adults who own smart speakers regarding how concerned they are about how much personal data their smart speakers collect. The results are shown in the table. Find a point estimate for the population proportion of U.S. adults who are very concerned. (Adapted from Speakergy) Answer: Page A40 EXAMPLE 1 How concerned are you? Number responding yes Very concerned 147 Somewhat concerned 420 Not too concerned 378 Not at all concerned 105 Study Tip In Sections 6.1 and 6.2, estimates were made for quantitative data. In this section, sample proportions are used to make estimates for qualitative data.
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