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4.4.2 Normal Approximations to Binomial Distributions

4.4.2 Normal Approximations to Binomial Distributions. Confidence intervals and sample size for proportions. Margin of error. Confidence level 95% probability that x is somewhere in the range ( x – 3.7, x+ 3.7). Confidence interval. Confidence Intervals, a review.

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4.4.2 Normal Approximations to Binomial Distributions

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  1. 4.4.2 Normal Approximations to Binomial Distributions Confidence intervals and sample size for proportions

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

  3. 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

  4. is the proportion in the sample 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

  5. Common confidence levels and their associated z-scores

  6. 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

  7. 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.

  8. 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

  9. 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

  10. 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

  11. Sample Size, n

  12. 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.

  13. Example 2 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.

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