1. Chapter 10: Basics of Confidence Intervals
February 19
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

2. 10: Intro to Confidence Intervals
Chapter 10
2/18/2019
In Chapter 10:
10.1 Introduction to Estimation
10.2 Confidence Interval for μ (σ known)
10.3 Sample Size Requirements
10.4 Relationship Between Hypothesis Testing and Confidence Intervals
2/18/2019Basic Biostat
10: Intro to Confidence Intervals
Basic Biostat

3. §10.1: Introduction to Estimation
Two forms of estimation
Point estimation ≡ most likely value of parameter (e.g., x-bar is point estimator of µ)
Interval estimation ≡ range of values with known likelihood of capturing the parameter, i.e., a confidence interval (CI)
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

4. 10: Intro to Confidence Intervals
Reasoning Behind a 95% CI
The next slide demonstrates how CIs are based on sampling distributions
If we take multiple samples from the sample population, each sample will derive a different 95% CI
95% of the CIs will capture μ & 5% will not
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

5. 10: Intro to Confidence Intervals
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

6. Confidence Interval for μ
To create a 95% confidence interval for μ, surround each sample mean with margin of error m: m ≈ 2×SE = 2×(σ/√n)
The 95% confidence interval for μ is:
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

7. 10: Intro to Confidence Intervals
Sampling distribution of a mean (curve).
Below the curve are five CIs.
In this example, all but the third CI captured μ
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

8. 10: Intro to Confidence Intervals
“Body Weight” Example
Body weights of year-old males have unknown μ and σ = 40
Take an SRS of n = 712 from population
Calculate: x-bar =183
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

9. Confidence Interval Formula
Here is a more accurate and flexible formula
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

10. Common Levels of Confidence
Confidence level
1 – α
Alpha level α
Z value
z1–(α/2)
.90
.10
1.645
.95
.05
1.960
.99
.01
2.576
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

11. 90% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

12. 95% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

13. 99% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

14. Confidence Level and CI Length
UCL ≡ Upper Confidence Limit; LCL ≡ Lower Limit;
Confidence level
Body weight example
CI length = UCL – LCL
90%
180.5 to 185.5
185.5 – = 5.0
95%
180.1 to 185.9
185.9 – = 5.8
99%
179.1 to 186.9
186.9 – = 7.8
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10: Intro to Confidence Intervals

15. 10.3 Sample Size Requirements
Ask: How large a sample is need to determine a (1 – α)100% CI with margin of error m?
Illustrative example: Recall that WAIS has σ = 15. Suppose we want a 95% CI for μ
For 95% confidence, α = .05, z1–.05/2 = z.975 = (Continued on next slide)
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

16. Illustrative Examples: Sample Size
Round up to ensure precision
Smaller m require larger n
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

17. 10.4 Relation Between Testing and Confidence Intervals
Rule: Rejects H0 at α level of significance when μ0 falls outside the (1−α)100% CI.
Illustration: Next slide
2/18/2019Basic Biostat
10: Intro to Confidence Intervals

18. Example: Testing and CIs
Illustration: Test H0: μ = 180
This CI excludes 180
Reject H0 at α =.05
Retain H0 at α =.01
This CI includes 180
2/18/2019Basic Biostat
10: Intro to Confidence Intervals