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1. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 1
§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. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 2
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

3. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 3

4. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 4
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:
mx

5. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 5
Sampling
distribution of a
mean (curve).
Below the curve
are five CIs.
In this example,
all but the third CI
captured μ

6. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 6
“Body Weight” Example
•Body weights of 20-29-year-old males
have unknown μ and σ = 40
•Take an SRS of n = 712 from population
•Calculate: x-bar =183
35.122 and 5.1
712
40

xx SEm
n
SE

pounds 186 to180
3183
for CI 95%


mx

7. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 7
Confidence Interval Formula
Here is a more accurate and flexible formula
xSEzx
n
zx




2
2
1
1
ly,Equivalent 



8. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 8
Confidence level
1 – α
Alpha level
α
Z value
z
1–(α/2)
.90 .10 1.645
.95 .05 1.960
.99 .01 2.576
Common Levels of Confidence

9. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 9
90% Confidence Interval for μ
5.185 to5.180
5.2183
712
40
645.1183
for CI %90
2
1.
1





n
zx


Data: SRS, n = 712, σ = 40, x-bar = 183

10. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 10
95% Confidence Interval for μ
9.185 to1.180
9.2183
712
40
960.1183
for CI %95
2
05.
1





n
zx


Data: SRS, n = 712, σ = 40, x-bar = 183

11. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 11
99% Confidence Interval for μ
9.186 to1.179
9.3183
712
40
576.2183
for CI %99
2
01.
1





n
zx


Data: SRS, n = 712, σ = 40, x-bar = 183

12. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 12
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 185.5185.5 – 180.5 = 5.0
95% 180.1 to 185.9185.9 – 180.1 = 5.8
99% 179.1 to 186.9186.9 – 179.1 = 7.8

13. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 13
10.3 Sample Size Requirements
2
1
2








m
zn


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, z
1–.05/2 = z
.975 = 1.96 (Continued on
next slide)

14. 09/18/24Basic Biostat 10: Intro to Confidence Intervals 14
Illustrative Examples: Sample Size
356.34
5
15
96.1 use ,5For
22
1
2














m
znm


(1) Round up to ensure precision
(2) Smaller m require larger n
1393.138
5.2
15
96.1 use ,5.2For
2






 nm
8654.864
1
15
96.1 use ,1For
2






 nm