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: of p ˆ

! Shape: Approximately Normal ! Center: The mean is p. ! Spread: The standard deviation is p(1− p) n

2 95% (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.

! : 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.

! : 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/

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