Confidence Intervals Chapter 6

§ 6.3 Confidence Intervals for Population Proportions

Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 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 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 = 1 – The symbols and are read as “p hat” and “q hat.”

Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 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. The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%. n = 1250x = 450

Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Confidence Intervals for p A c - confidence interval for the population proportion p is where 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. Continued. n = 1250x = 450

Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Confidence Intervals for p Example continued: 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%. Left endpoint = ?Right endpoint = ? n = 1250x = 450

Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 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 3.Verify that the sampling distribution can be approximated by the normal distribution. 4.Find the critical value z c 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. Use the Standard Normal Table. Left endpoint : Right endpoint : Interval :

Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Sample Size Given a c - confidence level and a margin of error, E, the minimum sample size n, needed to estimate p is This formula assumes you have an estimate for and If not, use and 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 9 Sample Size You should sample at least 2213 adults to be 95% confident. (Always round up.) 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 = 1250x = 450