4.4.2 Normal Approximations to Binomial Distributions Confidence intervals and sample size for proportions

Confidence Intervals, a review Margin of error These results are accurate to within +/- 3.7%, 19 times out of 20. Confidence level 95% probability that x is somewhere in the range (x – 3.7, x+3.7) Confidence interval

Confidence Intervals, a review A (1 – ) or (1 – ) x 100% confidence interval for , given population standard deviation , sample size n, and sample mean , represents the range of values

Confidence Levels for Proportions Often want to know proportion of population that have particular opinion or characteristic Proportion is p, the probability of success in binomial distribution is the proportion in the sample

Common confidence levels and their associated z-scores Tail size, z-score, 90% 0.05 1.645 95% 0.025 1.960 99% 0.005 2.576

Comments The sample proportion is an estimate of the population proportion (like sample mean is an estimate of the population mean) For many polls, population proportion not known In fact, purpose of poll is to estimate it! Can estimate using sample proportion

Example Voter turnout in municipal elections is often very low. In a recent election, the mayor got 53% of the voters, but only about 1500 voters turned out. Construct a 90% confidence interval for the proportion of people who support the mayor. Comment on any assumptions you have to make for your calculation.

Example 1a, sol’n Use election results to estimate proportion: These estimated values give a 90% confidence interval of The mayor can be 90% confident of having the support of 51-55% of the population

Example 1b, sol’n Have to assume that people who voted are representative of whole population Assumption might not be valid people who take trouble to vote likely to be the ones most interested in municipal affairs

Sample Sizes & Margin of Error Margin of Error = half the confidence interval width the maximum difference between the observed proportion in the sample and the true value of the proportion in the population w = 2E

Sample Size, n

Example 2 A recent survey indicated that 82% of secondary-school students graduate within five years of entering grade 9. This result is considered accurate within plus or minus 3%, 19 times out of 20. Estimate the sample size in this survey.

Example 2 You would need to sample about 630 students. For a 95% confidence level, z0.975 = 1.960 w = 2E = 2(3%) = 6% Use survey results to estimate p: You would need to sample about 630 students.