1 / 32

Chapter 11: Estimation

Chapter 11: Estimation. Motivating Example. Research Question : What proportion of all currently-housed U.S. adults ever experienced homelessness? Research Study* Random sample of 1,507 currently-housed adults in the U.S. Proportion of the sample who ever experienced homeless was 0.14.

keiki
Download Presentation

Chapter 11: Estimation

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. Chapter 11: Estimation

  2. Motivating Example • Research Question: What proportion of all currently-housed U.S. adults ever experienced homelessness? • Research Study* • Random sample of 1,507 currently-housed adults in the U.S. • Proportion of the sample who ever experienced homeless was 0.14 *Link, B. et al. (1994). “Lifetime and five-year prevalence of homelessness in the United States.” American Journal of Public Health, 84, 1907-1912.

  3. Motivating Example • Question 1: What is our best guess about the proportion of all currently-housed U.S. adults who ever experienced homelessness? • Answer: 0.14 • Comment: This is called a point estimate • Question 2: How good is our guess? • Answer: We are fairly sure that the true proportion is between 0.13 and 0.15 • Comment: This is called an interval estimate

  4. Estimation • Estimation: Using a sample statistic to estimate a population parameter • This is our ultimate goal in statistics! • Point Estimate: A sample statistic that is used to estimate the value of a population parameter • It is our “best guess” as to what’s going on in the population • Interval Estimate (Confidence Interval): A range of values within which the population parameter may fall • This gives us an idea about the accuracy of our point estimate

  5. Confidence Interval • Piece 1: Point estimate • We can use proportions or means • Calculated from the sample • You will be given this in class • Piece 2: Standard error of the point estimate • Calculated from the sample • You will have to calculate this • Piece 3: Confidence level • Defined based on a z-statistic from the normal distribution

  6. Confidence Level • Definition: The likelihood that a given confidence interval will contain the population parameter • Example: 95% Confidence Level • We are 95% confident that a specific interval contains the population parameter

  7. Confidence Level • Z-Statistic: We use a z-statistic from the normal distribution to define the confidence level • This is true when N > 50 • We are applying the central limit theorem! • Common Confidence Levels:

  8. Point Estimate: Sample proportion (p) This is an estimate of the population proportion (π) Standard Error: Z-Statistic: See Slide 7 Confidence Interval for a Proportion

  9. Confidence Interval: Mathematical Formulas Lower Limit: p – (Z·SE) Upper Limit: p + (Z·SE) Confidence Interval: Pictorial Representation Confidence Interval for a Proportion

  10. Example: From Slide 2 Sample Size: N = 1,507 Sample Proportion: p = 0.14 Standard Error: Confidence Interval for a Proportion

  11. 90% Confidence Interval: Interpretation: We are 90% confident that the proportion of all currently-housed U.S. adults who ever experienced homelessness is between 0.13 and 0.15 Confidence Interval for a Proportion I’m the Point Estimate Lower Limit Upper Limit

  12. Confidence Interval: 0 to 1 Reasoning: The only way we can be 100% confident is by considering every possible value from 0 to 1 100% Confidence Interval for a Proportion

  13. Confidence Interval for a Proportion: Factors Affecting Width • Sample Size • Effect: As the sample size increases, the confidence interval gets smaller (more precise) • This is holding the proportion and confidence level constant • Why? The standard error (SE) decreases as the sample size increases • Examples: See diagram on the next slide

  14. Confidence Interval for a Proportion: Factors Affecting Width

  15. Confidence Interval for a Proportion: Factors Affecting Width • Level of Confidence • Effect: As the level of confidence increases, the confidence interval gets larger (less precise) • This is holding the proportion and sample size constant • Why? The Z-statistic increases as the level of confidence increases • Examples: See diagram on the next slide

  16. Confidence Interval for a Proportion: Factors Affecting Width

  17. MoE: In the news, you will often see poll results and a “margin of error” Calculation: It includes the standard error assuming p = 0.50 It is based on a z-statistic of 1.96 (rounded up to 2) Derivation: For the math-geek types Margin of Error (MoE)

  18. Use: Construct a 95% confidence interval from the point estimate and the MoE 95% Confidence Interval: Mathematical Formulas Lower Limit: p – MoE Upper Limit: p + MoE 95% Confidence Interval: Pictorial Representation Margin of Error (MoE)

  19. Situation: In a clinical trial for Rozerem (a sleep aid), 6% of the 1,250 participants experienced dizziness Margin of Error: Margin of Error (MoE) Example:Dizziness From Rozerem

  20. 95% Confidence Interval: Interpretation: We are 95% confident that, among all people who take Rozerem, the proportion who will experience dizziness is between 0.03 and 0.09 Margin of Error (MoE) Example:Dizziness From Rozerem

  21. Point Estimate: Sample mean ( ) This is an estimate of the population mean (μ) Standard Error: Z-Statistic: See Slide 7 Confidence Interval for a Mean

  22. Confidence Interval: Mathematical Formulas Lower Limit: – (Z·SE) Upper Limit: + (Z·SE) Confidence Interval: Pictorial Representation Confidence Interval for a Mean

  23. Research Question: On average, how many hours a day do all Texas children ages 2-18 spend watching TV? Sample: From a sample of N = 749 children, the mean hours spent watching TV was with a standard deviation of S = 2.97 Example of Confidence Interval for a Mean: TV Watching

  24. Goal: Calculate and interpret a 99% confidence interval Standard Error: Example of Confidence Interval for a Mean: TV Watching

  25. 99% Confidence Interval: Interpretation: We are 99% confident that the average time spent watching TV among all Texas children ages 2-18 is between 3.26 and 3.82 hours Example of Confidence Interval for a Mean: TV Watching I’m the Point Estimate Lower Limit Upper Limit

  26. Confidence Interval: -∞ to +∞ Reasoning: The only way we can be 100% confident is by considering every possible value from -∞ to +∞ 100% Confidence Interval for a Mean

  27. Confidence Interval for a Mean: Factors Affecting Width • Sample Size • Effect: As the sample size increases, the confidence interval gets smaller (more precise) • This is holding the mean, standard deviation, and confidence level constant • Why? The standard error (SE) decreases as the sample size increases • Examples: See diagram on the next slide

  28. Confidence Interval for a Mean: Factors Affecting Width

  29. Confidence Interval for a Mean: Factors Affecting Width • Standard Deviation • Effect: As the standard deviation increases, the confidence interval gets larger (less precise) • This is holding the mean, sample size, and confidence level constant • Why? The standard error (SE) increases as the standard deviation increases • Examples: See diagram on the next slide

  30. Confidence Interval for a Mean: Factors Affecting Width

  31. Confidence Interval for a Mean: Factors Affecting Width • Level of Confidence • Effect: As the level of confidence increases, the confidence interval gets larger (less precise) • This is holding the mean, standard deviation, and sample size constant • Why? The Z-statistic increases as the level of confidence increases • Examples: See diagram on the next slide

  32. Confidence Interval for a Mean: Factors Affecting Width

More Related