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8.3 Estimating Population Proportions ! We Use the Sample Proportion P ˆ As Our Estimate of the Population Proportion P

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8.3 Estimating Population Proportions ! We Use the Sample Proportion P ˆ As Our Estimate of the Population Proportion P 8.3 Estimating Population Proportions ! We use the sample proportion p ˆ as our estimate of the population proportion p. ! We should report some kind of ‘confidence’ about our estimate. Do we think it’s pretty accurate? Or not so accurate. ! What sample size n do we need for a given level of confidence about our estimate. " Larger n coincides with better estimate. 1 Recall: Sampling Distribution of p ˆ ! Shape: Approximately Normal ! Center: The mean is p. ! Spread: The standard deviation is p(1− p) n 2 95% Confidence Interval (CI) for a Population Proportion p ! The margin of error (MOE) for the 95% CI for p is pˆ(1− pˆ) MOE = E ≈ 2 n where p ˆ is the sample proportion. That is, the 95% confidence interval ranges from ( p ˆ – margin of error) to ( p ˆ + margin of error). 3 95% Confidence Interval (CI) for a Population Proportion p ! We can write this confidence interval more formally as pˆ − E < p < pˆ + E Or more briefly as pˆ ± E 4 Example ! Population: Registered voters in the U.S. ! Parameter: Proportion of U.S. voters who are concerned about the spread of bird flu in the United States. Unknown! 5 Example ! Sample: 900 randomly selected registered voters nationwide. FOX News/Opinion Dynamics Poll, Oct. 11-12, 2005. ! Statistic: 63% of the sample are somewhat or very concerned about the spread of bird flu in the United States. 6 Confidence Interval for p ! We are 95% confident that p will fall between pˆ(1− pˆ) pˆ(1− pˆ) pˆ − 2 and pˆ + 2 n n MOE MOE 7 Example Best guess for p. pˆ = 0.63 pˆ(1− pˆ) 0.63(0.37) = = 0.016 n 900 0.63 − 2(0.016) to 0.63 + 2(0.016) MOE MOE 8 Example (cont.) ! We are 95% confident that p will fall between 0.63 − 2(0.016) = 0.598 and 0.63 + 2(0.016) = 0.662 or that it will be in the interval (0.598, 0.662) 9 Interpretation pˆ(1− pˆ) pˆ(1− pˆ) pˆ − 2 = 0.598 and pˆ + 2 = 0.662 n n Lower end Upper end ! We are 95% confident that the population proportion of registered voters in the U.S. who are concerned about the spread of bird flu in the U.S. is between 59.8% and 66.2%. 10 95% Confidence ! If one were to repeatedly sample at random 900 registered voters and compute a 95% confidence interval for each sample, 95% of the intervals produced would contain the population proportion p. " If we repeated the process 100 times, we expect that 95 of our confidence intervals would contain p. " If we repeated the process 100 times, we expect that 5 of our confidence intervals would NOT contain p. 11 Interpretation of the 95% Confidence Interval (CI) for a Population Proportion p ! We are 95% confident that this interval contains the true parameter value p. " Note that a 95% CI always contains pˆ . In fact, it’s right at the center of every 95% CI. " I might’ve missed the p with this interval, but at least I’ve set it up so that’s not very likely. 12 Interpretation of the 95% Confidence Interval (CI) for a Population Proportion p ! If I was to repeat this process 100 times (i.e. take a new sample, compute the CI, do again, etc.), then on average, 95 of those confidence intervals I created will contain p. " See applet linked at our website: http://statweb.calpoly.edu/chance/applets/ ConfSim/ConfSim.html 13 14 .
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