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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.

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

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

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

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

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:

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 ProportionConfidence Interval: Mathematical Formulas

Lower Limit: p – (Z·SE)

Upper Limit: p + (Z·SE)

Confidence Interval: Pictorial Representation

Confidence Interval for a ProportionExample: From Slide 2

Sample Size: N = 1,507

Sample Proportion: p = 0.14

Standard Error:

Confidence Interval for a ProportionInterpretation: 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 ProportionI’m the Point Estimate

Lower Limit

Upper Limit

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 ProportionConfidence 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

- Effect: As the sample size increases, the confidence interval gets smaller (more precise)

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

- Effect: As the level of confidence increases, the confidence interval gets larger (less precise)

MoE Width: 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)Use Width: 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)Situation Width: 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 Rozerem95% Confidence Interval Width:

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 RozeremPoint Estimate Width: Sample mean ( )

This is an estimate of the population mean (μ)

Standard Error:

Z-Statistic: See Slide 7

Confidence Interval for a MeanConfidence Interval Width: Mathematical Formulas

Lower Limit: – (Z·SE)

Upper Limit: + (Z·SE)

Confidence Interval: Pictorial Representation

Confidence Interval for a MeanResearch Question Width: 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 WatchingGoal Width: Calculate and interpret a 99% confidence interval

Standard Error:

Example of Confidence Interval for a Mean: TV Watching99% Confidence Interval Width:

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 WatchingI’m the Point Estimate

Lower Limit

Upper Limit

Confidence Interval Width: -∞ to +∞

Reasoning: The only way we can be 100% confident is by considering every possible value from -∞ to +∞

100% Confidence Interval for a MeanConfidence Interval for a Mean: Factors Affecting Width 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

- Effect: As the sample size increases, the confidence interval gets smaller (more precise)

Confidence Interval for a Mean: Factors Affecting Width 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

- Effect: As the standard deviation increases, the confidence interval gets larger (less precise)

Confidence Interval for a Mean: WidthFactors 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

- Effect: As the level of confidence increases, the confidence interval gets larger (less precise)

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