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STA 291 Fall 2009

STA 291 Fall 2009. Lecture 15 Dustin Lueker. Confidence Intervals. To calculate the confidence interval, we use the Central Limit Theorem ( np and nq ≥ 5) Also, we need a that is determined by the confidence level Formula for 100(1- α )% confidence interval for μ.

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STA 291 Fall 2009

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  1. STA 291Fall 2009 Lecture 15 Dustin Lueker

  2. Confidence Intervals • To calculate the confidence interval, we use the Central Limit Theorem (np and nq ≥ 5) • Also, we need a that is determined by the confidence level • Formula for 100(1-α)% confidence interval for μ STA 291 Fall 2009 Lecture 15

  3. Interpreting Confidence Intervals • Incorrect statement • With 95% probability, the population mean will fall in the interval from 3.5 to 5.2 • To avoid the misleading word “probability” we say • We are 95% confident that the true population mean will fall between 3.5 and 5.2 STA 291 Fall 2009 Lecture 15

  4. Confidence Interval • Changing our confidence level will change our confidence interval • Increasing our confidence level will increase the length of the confidence interval • A confidence level of 100% would require a confidence interval of infinite length • Not informative • There is a tradeoff between length and accuracy • Ideally we would like a short interval with high accuracy (high confidence level) STA 291 Fall 2009 Lecture 15

  5. Facts about Confidence Intervals • The width of a confidence interval • as the confidence level increases • as the error probability decreases • as the standard error increases • as the sample size n decreases • Why? STA 291 Fall 2009 Lecture 15

  6. Choice of Sample Size • Start with the confidence interval formula for a population proportion p • ME denotes the margin of error • Mathematically we need to solve the above equation for n STA 291 Fall 2009 Lecture 15

  7. Choice of Sample Size • This formula requires guessing before taking the sample, or taking the safe but conservative approach of letting = .5 • Why is this the worst case scenario? (conservative approach) STA 291 Fall 2009 Lecture 15

  8. Example • ABC/Washington Post poll (December 2006) • Sample size of 1005 • Question • Do you approve or disapprove of the way George W. Bush is handling his job as president? • 362 people approved • Construct a 95% confidence interval for p • What is the margin of error? STA 291 Fall 2009 Lecture 15

  9. Example • If we wanted B=2%, using the sample proportion from the Washington Post poll, recall that the sample proportion was .36 • n=2212.7, so we need a sample of 2213 • What do we get if we use the conservative approach? STA 291 Fall 2009 Lecture 15

  10. Confidence Interval for Unknown σ • To account for the extra variability of using a sample size of less than 30 the student’s t-distribution is used instead of the normal distribution STA 291 Fall 2009 Lecture 15

  11. t-distribution • t-distributions are bell-shaped and symmetric around zero • The smaller the degrees of freedom the more spread out the distribution is • t-distribution look much like normal distributions • In face, the limit of the t-distribution is a normal distribution as n gets larger STA 291 Fall 2009 Lecture 15

  12. Finding tα/2 • Need to know α and degrees of freedom (df) • df = n-1 • α=.05, n=23 • tα/2= • α=.01, n=17 • tα/2= • α=.1, n=20 • tα/2= STA 291 Fall 2009 Lecture 15

  13. Example • Compute a 95% confidence interval for μ if we know that s=12 and the sample of size 36 yielded a mean of 7 STA 291 Fall 2009 Lecture 15

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