Basics of Confidence Intervals

2/20/2023Basic Biostat 10: Intro to Confidence Intervals 1
February 23
Chapter 10:
Basics of Confidence Intervals

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

§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)

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

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:
m
x 

Sampling
distribution of a
mean (curve).
Below the curve
are five CIs.
In this example,
all but the third CI
captured μ

“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
3
5
.
1
2
2
and
5
.
1
712
40







 x
x SE
m
n
SE

pounds
186
to
180
3
183
for
CI
95%




 m
x


Confidence Interval Formula
Here is a more accurate and flexible formula
x
SE
z
x
n
z
x






2
2
1
1
ly,
Equivalent 



Confidence level
1 – α
Alpha level
α
Z value
z1–(α/2)
.90 .10 1.645
.95 .05 1.960
.99 .01 2.576
Common Levels of Confidence

90% Confidence Interval for μ
5
.
185
to
5
.
180
5
.
2
183
712
40
645
.
1
183
for
CI
%
90
2
1
.
1








 
n
z
x


Data: SRS, n = 712, σ = 40, x-bar = 183

9
.
185
to
1
.
180
9
.
2
183
712
40
960
.
1
183
for
CI
%
95
2
05
.
1








 
n
z
x


Data: SRS, n = 712, σ = 40, x-bar = 183

9
.
186
to
1
.
179
9
.
3
183
712
40
576
.
2
183
for
CI
%
99
2
01
.
1








 
n
z
x


Data: SRS, n = 712, σ = 40, x-bar = 183

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 – 180.5 = 5.0
95% 180.1 to 185.9 185.9 – 180.1 = 5.8
99% 179.1 to 186.9 186.9 – 179.1 = 7.8

10.3 Sample Size Requirements
2
1 2







 
m
z
n


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 = 1.96
(Continued on next slide)

Illustrative Examples: Sample Size
35
6
.
34
5
15
96
.
1
use
,
5
For
2
2
1 2


















 
m
z
n
m


(1) Round up to ensure precision
(2) Smaller m require larger n
139
3
.
138
5
.
2
15
96
.
1
use
,
5
.
2
For
2










 n
m
865
4
.
864
1
15
96
.
1
use
,
1
For
2










 n
m

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

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

Basics of Confidence Intervals

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