1. Chapter 10: Basics of Confidence Intervals
April 17

2. In Chapter 10: 10.1 Introduction to Estimation
4/24/2017
In Chapter 10:
10.1 Introduction to Estimation
10.2 Confidence Interval for μ when σ is known
10.3 Sample Size Requirements
10.4 Relationship Between Hypothesis Testing and Confidence Intervals
Basic Biostat

3. §10.1: Introduction to Estimation
Two forms of estimation
Point estimation ≡ single best estimate of parameter (e.g., x-bar is the point estimate of μ)
Interval estimation ≡ surrounding the point estimate with a margin of error to create a range of values that seeks to capture the parameter; a confidence interval

4. Reasoning Behind a 95% Confidence Interval
A schematic (next slide) of a sampling distribution of means based on repeated independent SRSs of n = 712 is taken from a population with unknown μ and σ = 40.
Each sample derives a different point estimate and 95% confidence interval
95% of the confidence intervals will capture the value of μ

6. Confidence Intervals
To create a 95% confidence interval for μ, surround each sample mean with a margin of error m that is equal to 2standard errors of the mean: m ≈ 2×SE = 2×(σ/√n)
The 95% confidence interval for μ is now

7. This figure shows a sampling distribution of means.
Below the sampling distribution are five confidence intervals.
In this instance, all but the third confidence captured μ

8. Example: Rough Confidence Interval
Suppose body weights of year-old males has unknown μ and σ = 40.
I take an SRS of n = 712 from this population and calculate x-bar =183. Thus:

9. Confidence Interval Formula
Here is a better formula for a (1−α)100% confidence interval for μ when σ is known:
Note that σ/√n is the SE of the mean

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

11. 90% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183

12. 95% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183

13. 99% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183

14. Confidence Level and CI Length
↑ confidence costs  ↑ confidence interval length
Confidence level
Illustrative CI
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

15. 10.3 Sample Size Requirements
To derive a confidence interval for μ with margin of error m, study this many individuals:

16. Examples: Sample Size Requirements
Suppose we have a variable with s = 15 and want a 95% confidence interval. Note, α = .05  z1–.05/2 = z.975 = 1.96
round up to ensure precision
Smaller margins of error require larger sample sizes

17. 10.4 Relationship Between Hypothesis Testing and Confidence Intervals
A two-sided test will reject the null hypothesis at the α level of significance when the value of μ0 falls outside the (1−α)100% confidence interval
This illustration rejects H0: μ = 180 at α =.05 because 180 falls outside the 95% confidence interval.
It retains H0: μ = 180 at α = .01 because the 99% confidence interval captures 180.