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Apr 20, 2023
Chapter 10: Chapter 10: Basics of Confidence IntervalsBasics of Confidence Intervals
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 EstimationTwo 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
mx
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:
pounds 186 to1803183for CI 95%
35.122
5.1712
40
mx
SEmn
SE
x
x
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
nzx
21
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 to5.180
5.2183712
40645.1183
for CI %9021.1
n
zx
Data: SRS, n = 712, σ = 40, x-bar = 183
95% Confidence Interval for μ
9.185 to1.180
9.2183712
40960.1183
for CI %95205.1
n
zx
Data: SRS, n = 712, σ = 40, x-bar = 183
99% Confidence Interval for μ
9.186 to1.179
9.3183712
40576.2183
for CI %99201.1
n
zx
Data: SRS, n = 712, σ = 40, x-bar = 183
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 – 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
zn
To derive a confidence interval for μ with margin of error m, study this many individuals:
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
356.345
1596.1 use ,5For
22
1 2
m
znm
Smaller margins of error require larger sample sizes
1393.1385.2
1596.1 use ,5.2For
2
nm
8654.8641
1596.1 use ,1For
2
nm
round up to ensure precision
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.
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