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Confidence Intervals and Interpretation

Learn about confidence intervals, their interpretation, the choice of sample size, facts about confidence intervals, t-distribution, finding tα/2, examples of confidence intervals for means and proportions, and sample size determination.

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Confidence Intervals and Interpretation

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  1. STA 291Spring 2010 Lecture 14 Dustin Lueker

  2. Confidence Intervals • This interval will contain μ with a 100(1-α)% confidence • If we are estimating µ, then why it is unreasonable for us to know σ? • Thus we replace σ by s (sample standard deviation) • This formula is used for a large sample size (n≥30) • If we have a sample size less than 30 a different distribution is used, the t-distribution, we will get to this later STA 291 Spring 2010 Lecture 14

  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 Spring 2010 Lecture 14

  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 Spring 2010 Lecture 14

  5. Choice of Sample Size • Start with the confidence interval formula assuming that the population standard deviation is known • Mathematically we need to solve the above equation for n STA 291 Spring 2010 Lecture 14

  6. Facts about Confidence Intervals • The width of a confidence interval • as the confidence level increases • as the error probability increases • as the standard error increases • as the sample size n increases • Why? STA 291 Spring 2010 Lecture 14

  7. Confidence Interval for µ with small n • 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 Spring 2010 Lecture 14

  8. 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 Spring 2010 Lecture 14

  9. 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 Spring 2010 Lecture 14

  10. Example • A sample of 12 individuals yields a mean of 5.4 and a variance of 16. Estimate the population mean with 98% confidence. STA 291 Spring 2010 Lecture 14

  11. Confidence Interval for a Proportion • The sample proportion is an unbiased and efficient estimator of the population proportion • The proportion is a special case of the mean STA 291 Spring 2010 Lecture 14

  12. 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 Spring 2010 Lecture 14

  13. Sample Size • As with a confidence interval for the sample mean a desired sample size for a given margin of error (E) and confidence level can be computed for a confidence interval about the sample proportion • 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 Spring 2010 Lecture 14

  14. 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 Spring 2010 Lecture 14

  15. Confidence Interval for p • To calculate the confidence interval, we use the Central Limit Theorem (np and nq ≥ 5) • What if this isn’t satisfied? • Instead of the typical estimator, we will use • Then the formula for confidence interval becomes STA 291 Spring 2010 Lecture 14

  16. Example • Suppose a student in an advertising class is studying the impact of ads placed during the Super Bowl, and wants to know what the proportion of students on campus watched it. She takes a random sample of 25 students and finds that all 25 watched the Super Bowl. • Find a 95% confidence interval using first method learned for p • Find a 95% confidence interval using the new method if np, nq condition fails STA 291 Spring 2010 Lecture 14

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