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Chapter 6 Confidence Intervals § 6.3 Confidence Intervals for Population Proportions Point Estimate for Population p The probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by pˆ x n where x is the number of successes in the sample and n is the number in the sample. The point estimate for the proportion of failures is qˆ = 1 – p̂. The symbols p̂ and qˆ are read as “p hat” and “q hat.” Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Point Estimate for Population p Example: In a survey of 1250 US adults, 450 of them said that their favorite sport to watch is baseball. Find a point estimate for the population proportion of US adults who say their favorite sport to watch is baseball. n = 1250 x = 450 pˆ x 450 0.36 n 1250 The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%. Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Confidence Intervals for p A c-confidence interval for the population proportion p is pˆ E p pˆ E where E zc pq ˆ ˆ. n The probability that the confidence interval contains p is c. Example: Construct a 90% confidence interval for the proportion of US adults who say baseball is their favorite sport to watch. n = 1250 x = 450 p̂ 0.36 Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Confidence Intervals for p Example continued: n = 1250 p̂ 0.36 x = 450 ˆˆ E z c pq n qˆ 0.64 Left endpoint = ? • p̂ E 0.36 0.022 0.338 1.645 (0.36)(0.64) 0.022 1250 Right endpoint = ? p̂ • 0.36 • p̂ E 0.36 0.022 0.382 With 90% confidence we can say that the proportion of all US adults who say baseball is their favorite sport to watch is between 33.8% and 38.2%. Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Finding Confidence Intervals for p 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 p̂. 3. Verify that the sampling distribution can be approximated by the normal distribution. 4. Find the critical value zc that corresponds to the given level of confidence. 5. Find the margin of error E. 6. Find the left and right endpoints and form the confidence interval. pˆ x n npˆ 5, nqˆ 5 Use the Standard Normal Table. ˆˆ E z c pq n Left endpoint: p̂ E Right endpoint: p̂ E Interval: pˆ E p pˆ E Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 Sample Size Given a c-confidence level and a margin of error, E, the minimum sample size n, needed to estimate p is 2 zc ˆˆ . n pq E This formula assumes you have an estimate for p̂ and qˆ. If not, use pˆ 0.5 and qˆ 0.5. Example: You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Sample Size Example continued: You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. n = 1250 x = 450 2 p̂ 0.36 1.96 z ˆ ˆ c (0.36)(0.64) n pq 0.02 E 2 2212.8 (Always round up.) You should sample at least 2213 adults to be 95% confident. Larson & Farber, Elementary Statistics: Picturing the World, 3e 9