Transcript Chapter 10: Basics of Confidence Intervals

Chapter 10:
Basics of Confidence Intervals
March 16
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
§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
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
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
xm
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
Example: Rough Confidence
Interval
Suppose body weights of 20-29-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:
SEx 


40
 1.5
n
712
m  2  SEx  2 1.5  3
95% CI for   x  m  183  3  180 to 186 pounds
Confidence Interval Formula
Here is a better formula for a (1−α)100%
confidence interval for μ when σ is known:
x  z1  
2

n
Note that σ/√n is the SE of the mean
Common Levels of Confidence
Confidence level
1–α
.90
Alpha level
α
.10
Z value
z1–(α/2)
1.645
.95
.05
1.960
.99
.01
2.576
90% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183
90 % CI for   x  z1 .1 
2

n
 183  1.645 
40
712
 183  2.5
 180 .5 to 185 .5
95% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183
95 % CI for   x  z1 .0 5 
2

n
 183  1.960 
40
712
 183  2.9
 180 .1 to 185 .9
99% Confidence Interval for μ
Data: SRS, n = 712, σ = 40, x-bar = 183
99 % CI for   x  z1 .0 1 
2

n
 183  2.576 
40
712
 183  3.9
 179 .1 to 186 .9
Confidence Level and CI Length
↑ confidence costs  ↑ confidence interval length
Confidence
level
90%
95%
99%
Illustrative CI
CI length = UCL – LCL
180.5 to 185.5
180.1 to 185.9
185.5 – 180.5 = 5.0
185.9 – 180.1 = 5.8
179.1 to 186.9
186.9 – 179.1 = 7.8
10.3 Sample Size Requirements
To derive a confidence interval for μ with
margin of error m, study this many
individuals:


n   z1   
 2 m
2
Examples: Sample Size Requirements
Suppose we have a variable with  = 15 and want a 95%
confidence interval. Note, α = .05  z1–.05/2 = z.975 = 1.96

For m  5, use n   z1 
 2

2
2
15 

   1.96    34 .6  35
m
5

2
round up to ensure precision
15 

For m  2.5, use n  1.96 
  138 .3  139
2.5 

2
15 

For m  1, use n  1.96    864 .4  865
1

Smaller margins of error require larger sample sizes
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.