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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.

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Chapter 11: Estimation

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  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

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