1 / 13

Understanding Confidence Intervals: Concepts and Calculations

This lecture explores the concept of confidence intervals, highlighting the Central Limit Theorem's role in their calculation. It covers how different confidence levels impact interval width and accuracy, as well as the significance of sample size in determining margin of error. Key misunderstandings are clarified, such as the proper interpretation of confidence intervals and the relationship between confidence level and interval length. Examples include real polls and calculations required for effective statistical analysis. Gain insight into using t-distributions for small sample sizes to ensure reliable estimates.

soren
Télécharger la présentation

Understanding Confidence Intervals: Concepts and Calculations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related