last time central limit theorem –illustrations –how large n? –normal approximation to binomial...
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Last Time
• Central Limit Theorem– Illustrations– How large n?– Normal Approximation to Binomial
• Statistical Inference– Estimate unknown parameters– Unbiasedness (centered correctly)– Standard error (measures spread)
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Administrative Matters
Midterm II, coming Tuesday, April 6
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Administrative Matters
Midterm II, coming Tuesday, April 6
• Numerical answers:– No computers, no calculators
![Page 4: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/4.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Numerical answers:– No computers, no calculators– Handwrite Excel formulas (e.g. =9+4^2)– Don’t do arithmetic (e.g. use such formulas)
![Page 5: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/5.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Numerical answers:– No computers, no calculators– Handwrite Excel formulas (e.g. =9+4^2)– Don’t do arithmetic (e.g. use such formulas)
• Bring with you:– One 8.5 x 11 inch sheet of paper
![Page 6: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/6.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Numerical answers:– No computers, no calculators– Handwrite Excel formulas (e.g. =9+4^2)– Don’t do arithmetic (e.g. use such formulas)
• Bring with you:– One 8.5 x 11 inch sheet of paper– With your favorite info (formulas, Excel, etc.)
![Page 7: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/7.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Numerical answers:– No computers, no calculators– Handwrite Excel formulas (e.g. =9+4^2)– Don’t do arithmetic (e.g. use such formulas)
• Bring with you:– One 8.5 x 11 inch sheet of paper– With your favorite info (formulas, Excel, etc.)
• Course in Concepts, not Memorization
![Page 8: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/8.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Material Covered:
HW 6 – HW 10
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Administrative Matters
Midterm II, coming Tuesday, April 6
• Material Covered:
HW 6 – HW 10
– Note: due Thursday, April 2
![Page 10: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/10.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Material Covered:
HW 6 – HW 10
– Note: due Thursday, April 2– Will ask grader to return Mon. April 5– Can pickup in my office (Hanes 352)
![Page 11: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/11.jpg)
Administrative Matters
Midterm II, coming Tuesday, April 6
• Material Covered:
HW 6 – HW 10
– Note: due Thursday, April 2– Will ask grader to return Mon. April 5– Can pickup in my office (Hanes 352)– So today’s HW not included
![Page 12: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/12.jpg)
Administrative Matters
Extra Office Hours before Midterm II
Monday, Apr. 23 8:00 – 10:00
Monday, Apr. 23 11:00 – 2:00
Tuesday, Apr. 24 8:00 – 10:00
Tuesday, Apr. 24 1:00 – 2:00
(usual office hours)
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Study Suggestions
1. Work an Old Exama) On Blackboard
b) Course Information Section
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Study Suggestions
1. Work an Old Exama) On Blackboard
b) Course Information Section
c) Afterwards, check against given solutions
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Study Suggestions
1. Work an Old Exama) On Blackboard
b) Course Information Section
c) Afterwards, check against given solutions
2. Rework HW problems
![Page 16: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/16.jpg)
Study Suggestions
1. Work an Old Exama) On Blackboard
b) Course Information Section
c) Afterwards, check against given solutions
2. Rework HW problemsa) Print Assignment sheets
b) Choose problems in “random” order
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Study Suggestions
1. Work an Old Exama) On Blackboard
b) Course Information Section
c) Afterwards, check against given solutions
2. Rework HW problemsa) Print Assignment sheets
b) Choose problems in “random” order
c) Rework (don’t just “look over”)
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Reading In Textbook
Approximate Reading for Today’s Material:
Pages 356-369, 487-497
Approximate Reading for Next Class:
Pages 498-501, 418-422, 372-390
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Law of AveragesCase 2: any random sample
CAN SHOW, for n “large”
is “roughly”
Terminology: “Law of Averages, Part 2” “Central Limit Theorem”
(widely used name)
nXX ,,1
X ,N
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Central Limit TheoremIllustration: Rice Univ. Applethttp://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
Starting Distribut’n
user input
(very non-Normal)
Dist’n of average
of n = 25
(seems very
mound shaped?)
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Extreme Case of CLTConsequences:
roughly
roughly
Terminology: Called
The Normal Approximation to the Binomial
p npppN 1,
X pnpnpN 1,
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Normal Approx. to BinomialHow large n?
• Bigger is better
• Could use “n ≥ 30” rule from above
Law of Averages
• But clearly depends on p
• Textbook Rule:
OK when {np ≥ 10 & n(1-p) ≥ 10}
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Statistical InferenceIdea: Develop formal framework for
handling unknowns p & μ
e.g. 1: Political Polls
e.g. 2a: Population Modeling
e.g. 2b: Measurement Error
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Statistical InferenceA parameter is a numerical feature of
population, not sample
An estimate of a parameter is some function of data
(hopefully close to parameter)
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Statistical InferenceStandard Error: for an unbiased estimator,
standard error is standard deviation
Notes: For SE of , since don’t know p, use
sensible estimate For SE of , use sensible estimate
pp
s
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Statistical InferenceAnother view:
Form conclusions by
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Statistical InferenceAnother view:
Form conclusions by
quantifying uncertainty
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Statistical InferenceAnother view:
Form conclusions by
quantifying uncertainty
(will study several approaches, first is…)
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Confidence Intervals
Background:
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Confidence Intervals
Background:
The sample mean, , is an “estimate”
of the population mean,
X
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Confidence Intervals
Background:
The sample mean, , is an “estimate”
of the population mean,
How accurate?
X
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Confidence Intervals
Background:
The sample mean, , is an “estimate”
of the population mean,
How accurate?
(there is “variability”, how
much?)
X
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Confidence IntervalsIdea:
Since a point estimate
(e.g. or )X p
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Confidence IntervalsIdea:
Since a point estimate
is never exactly right
(in particular ) 0XP
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Confidence IntervalsIdea:
Since a point estimate
is never exactly right
give a reasonable range of likely values
(range also gives feeling
for accuracy of estimation)
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Confidence IntervalsIdea:
Since a point estimate
is never exactly right
give a reasonable range of likely values
(range also gives feeling
for accuracy of estimation)
![Page 37: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/37.jpg)
Confidence IntervalsE.g. ,~,,1 NXX n
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Confidence IntervalsE.g. with σ known ,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Think: measurement error
,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Think: measurement error
Each measurement is Normal
,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Think: measurement error
Each measurement is Normal
Known accuracy (maybe)
,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Think: population modeling
,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Think: population modeling
Normal population
,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Think: population modeling
Normal population
Known s.d.
(a stretch, really need to improve)
,~,,1 NXX n
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Confidence IntervalsE.g. with σ known
Recall the Sampling Distribution:
,~,,1 NXX n
nNX
,~
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Confidence IntervalsE.g. with σ known
Recall the Sampling Distribution:
(recall have this even when data not
normal, by Central Limit Theorem)
,~,,1 NXX n
nNX
,~
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Confidence IntervalsE.g. with σ known
Recall the Sampling Distribution:
Use to analyze variation
,~,,1 NXX n
nNX
,~
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Confidence Intervals
Understand error as:
(normal density quantifies
randomness in )
ndistX '
X
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Confidence Intervals
Understand error as:
(distribution centered at μ)
ndistX '
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Confidence Intervals
Understand error as:
(spread: s.d. = )
ndistX 'n
n
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Confidence Intervals
Understand error as:
How to explain to untrained consumers?
ndistX 'n
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Confidence Intervals
Understand error as:
How to explain to untrained consumers?
(who don’t know randomness,
distributions, normal curves)
ndistX 'n
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Confidence Intervals
Approach: present an interval
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Confidence Intervals
Approach: present an interval
With endpoints:
Estimate +- margin of error
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Confidence Intervals
Approach: present an interval
With endpoints:
Estimate +- margin of error
I.e. mX
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Confidence Intervals
Approach: present an interval
With endpoints:
Estimate +- margin of error
I.e.
reflecting variability
mX
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Confidence Intervals
Approach: present an interval
With endpoints:
Estimate +- margin of error
I.e.
reflecting variability
mX
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Confidence Intervals
Approach: present an interval
With endpoints:
Estimate +- margin of error
I.e.
reflecting variability
How to choose ?
mX
m
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Confidence Intervals
Choice of Confidence Interval Radius
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Confidence Intervals
Choice of Confidence Interval Radius,
i.e. margin of error, m
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Confidence Intervals
Choice of Confidence Interval Radius,
i.e. margin of error, :
Notes:
• No Absolute Range (i.e. including “everything”) is available
m
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Confidence Intervals
Choice of Confidence Interval Radius,
i.e. margin of error, :
Notes:
• No Absolute Range (i.e. including “everything”) is available
• From infinite tail of normal dist’n
m
![Page 63: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/63.jpg)
Confidence Intervals
Choice of Confidence Interval Radius,
i.e. margin of error, :
Notes:
• No Absolute Range (i.e. including “everything”) is available
• From infinite tail of normal dist’n
• So need to specify desired accuracy
m
![Page 64: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/64.jpg)
Confidence IntervalsChoice of margin of error, m
![Page 65: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/65.jpg)
Confidence IntervalsChoice of margin of error, :Approach:• Choose a Confidence Level
m
![Page 66: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/66.jpg)
Confidence IntervalsChoice of margin of error, :Approach:• Choose a Confidence Level• Often 0.95
m
![Page 67: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/67.jpg)
Confidence IntervalsChoice of margin of error, :Approach:• Choose a Confidence Level• Often 0.95
(e.g. FDA likes this number for approving new drugs, and it is a common standard for publication in many fields)
m
![Page 68: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/68.jpg)
Confidence IntervalsChoice of margin of error, :Approach:• Choose a Confidence Level• Often 0.95
(e.g. FDA likes this number for approving new drugs, and it is a common standard for publication in many fields)
• And take margin of error to include that part of sampling distribution
m
![Page 69: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/69.jpg)
Confidence Intervals
E.g. For confidence level 0.95, want
0.95 = Area
![Page 70: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/70.jpg)
Confidence Intervals
E.g. For confidence level 0.95, want
distribution
0.95 = Area
X
![Page 71: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/71.jpg)
Confidence Intervals
E.g. For confidence level 0.95, want
distribution
0.95 = Area
= margin of errorm
X
![Page 72: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/72.jpg)
Confidence Intervals
Computation: Recall NORMINV
![Page 73: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/73.jpg)
Confidence Intervals
Computation: Recall NORMINV takes
areas (probs)
![Page 74: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/74.jpg)
Confidence Intervals
Computation: Recall NORMINV takes
areas (probs), and returns cutoffs
![Page 75: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/75.jpg)
Confidence Intervals
Computation: Recall NORMINV takes
areas (probs), and returns cutoffs
Issue: NORMINV works with lower areas
![Page 76: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/76.jpg)
Confidence Intervals
Computation: Recall NORMINV takes
areas (probs), and returns cutoffs
Issue: NORMINV works with lower areas
Note: lower tail
included
![Page 77: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/77.jpg)
Confidence Intervals
So adapt needed probs to lower areas….
![Page 78: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/78.jpg)
Confidence Intervals
So adapt needed probs to lower areas….
When inner area = 0.95,
![Page 79: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/79.jpg)
Confidence Intervals
So adapt needed probs to lower areas….
When inner area = 0.95,
Right tail = 0.025
![Page 80: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/80.jpg)
Confidence Intervals
So adapt needed probs to lower areas….
When inner area = 0.95,
Right tail = 0.025
Shaded Area = 0.975
![Page 81: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/81.jpg)
Confidence Intervals
So adapt needed probs to lower areas….
When inner area = 0.95,
Right tail = 0.025
Shaded Area = 0.975
So need to compute as:
nNORMINV
,,975.0
![Page 82: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/82.jpg)
Confidence Intervals
Need to compute:
nNORMINV
,,975.0
![Page 83: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/83.jpg)
Confidence Intervals
Need to compute:
Major problem: is unknown
nNORMINV
,,975.0
![Page 84: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/84.jpg)
Confidence Intervals
Need to compute:
Major problem: is unknown
• But should answer depend on ?
nNORMINV
,,975.0
![Page 85: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/85.jpg)
Confidence Intervals
Need to compute:
Major problem: is unknown
• But should answer depend on ?
• “Accuracy” is only about spread
nNORMINV
,,975.0
![Page 86: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/86.jpg)
Confidence Intervals
Need to compute:
Major problem: is unknown
• But should answer depend on ?
• “Accuracy” is only about spread
• Not centerpoint
nNORMINV
,,975.0
![Page 87: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/87.jpg)
Confidence Intervals
Need to compute:
Major problem: is unknown
• But should answer depend on ?
• “Accuracy” is only about spread
• Not centerpoint
• Need another view of the problem
nNORMINV
,,975.0
![Page 88: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/88.jpg)
Confidence Intervals
Approach to unknown
![Page 89: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/89.jpg)
Confidence Intervals
Approach to unknown :
Recenter, i.e. look at dist’n
X
![Page 90: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/90.jpg)
Confidence Intervals
Approach to unknown :
Recenter, i.e. look at dist’n
X
![Page 91: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/91.jpg)
Confidence Intervals
Approach to unknown :
Recenter, i.e. look at dist’n
Key concept:
Centered at 0
X
![Page 92: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/92.jpg)
Confidence Intervals
Approach to unknown :
Recenter, i.e. look at dist’n
Key concept:
Centered at 0
Now can calculate as:
nNORMINVm
,0,975.0
X
![Page 93: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/93.jpg)
Confidence Intervals
Computation of:
nNORMINVm
,0,975.0
![Page 94: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/94.jpg)
Confidence Intervals
Computation of:
Smaller Problem: Don’t know
nNORMINVm
,0,975.0
![Page 95: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/95.jpg)
Confidence Intervals
Computation of:
Smaller Problem: Don’t know
Approach 1: Estimate with
(natural approach: use estimate)
nNORMINVm
,0,975.0
s
![Page 96: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/96.jpg)
Confidence Intervals
Computation of:
Smaller Problem: Don’t know
Approach 1: Estimate with
• Leads to complications
nNORMINVm
,0,975.0
s
![Page 97: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/97.jpg)
Confidence Intervals
Computation of:
Smaller Problem: Don’t know
Approach 1: Estimate with
• Leads to complications
• Will study later
nNORMINVm
,0,975.0
s
![Page 98: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/98.jpg)
Confidence Intervals
Computation of:
Smaller Problem: Don’t know
Approach 1: Estimate with
• Leads to complications
• Will study later
Approach 2: Sometimes know
nNORMINVm
,0,975.0
s
![Page 99: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/99.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
~1?
![Page 100: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/100.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
~2?
![Page 101: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/101.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
~7?
![Page 102: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/102.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
~10?
![Page 103: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/103.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
Early Approach:
Use data to choose
window width
![Page 104: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/104.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
Challenge:
Not enough info in
data for good choice
![Page 105: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/105.jpg)
Research Corner
How many bumps in stamps data?
Kernel Density Estimates
Depends on Window
Alternate Approach:
Scale Space
![Page 106: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/106.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
![Page 107: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/107.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
![Page 108: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/108.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
• Terminology from Computer Vision
![Page 109: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/109.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
• Terminology from Computer Vision
(goal: teach computers to “see”)
![Page 110: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/110.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
• Terminology from Computer Vision:– Oversmoothing: coarse scale view
(zoomed out – macroscopic perception)
![Page 111: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/111.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
• Terminology from Computer Vision:– Oversmoothing: coarse scale view
– Undersmoothing: fine scale view
(zoomed in – microscopic perception)
![Page 112: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/112.jpg)
Research Corner
Scale Space:
View 1: Rainbow colored movie
![Page 113: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/113.jpg)
Research Corner
Scale Space:
View 2: Rainbow
colored overlay
![Page 114: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/114.jpg)
Research Corner
Scale Space:
View 3: Rainbow
colored surface
![Page 115: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/115.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
Challenge: how to do statistical inference?
![Page 116: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/116.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
Challenge: how to do statistical inference?
Which bumps are really there?
![Page 117: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/117.jpg)
Research Corner
Scale Space:
Main Idea:
• Don’t try to choose window width
• Instead use all of them
Challenge: how to do statistical inference?
Which bumps are really there?
(i.e. statistically significant)
![Page 118: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/118.jpg)
Research Corner
Scale Space:
Challenge: how to do statistical inference?
Which bumps are really there?
(i.e. statistically significant)
![Page 119: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/119.jpg)
Research Corner
Scale Space:
Challenge: how to do statistical inference?
Which bumps are really there?
(i.e. statistically significant)
Address this next time
![Page 120: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/120.jpg)
Confidence Intervals
E.g. Crop researchers plant 15 plots
with a new variety of corn.
![Page 121: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/121.jpg)
Confidence Intervals
E.g. Crop researchers plant 15 plots
with a new variety of corn. The
yields, in bushels per acre are:
138
139.1
113
132.5
140.7
109.7
118.9
134.8
109.6
127.3
115.6
130.4
130.2
111.7
105.5
![Page 122: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/122.jpg)
Confidence Intervals
E.g. Crop researchers plant 15 plots
with a new variety of corn. The
yields, in bushels per acre are:
Assume that = 10 bushels / acre
138
139.1
113
132.5
140.7
109.7
118.9
134.8
109.6
127.3
115.6
130.4
130.2
111.7
105.5
![Page 123: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/123.jpg)
Confidence IntervalsE.g. Find:
a) The 90% Confidence Interval for the mean value , for this type of corn.
b) The 95% Confidence Interval.
c) The 99% Confidence Interval.
d) How do the CIs change as the confidence level increases?
Solution, part 1 of Class Example 11:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg11.xls
![Page 124: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/124.jpg)
Confidence IntervalsE.g. Find:
a) 90% Confidence
Interval for
Next study relevant parts of E.g. 11:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg11.xls
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Confidence IntervalsE.g. Find:
a) 90% Confidence
Interval for
Use Excel
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Confidence IntervalsE.g. Find:
a) 90% Confidence
Interval for
Use Excel
Data in C8:C22
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Confidence IntervalsE.g. Find:
a) 90% Confidence Interval for
Steps:
- Sample Size, n
![Page 128: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/128.jpg)
Confidence IntervalsE.g. Find:
a) 90% Confidence Interval for
Steps:
- Sample Size, n
- Average,
X
![Page 129: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/129.jpg)
Confidence IntervalsE.g. Find:
a) 90% Confidence Interval for
Steps:
- Sample Size, n
- Average,
- S. D., σ
X
![Page 130: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/130.jpg)
Confidence IntervalsE.g. Find:
a) 90% Confidence Interval for
Steps:
- Sample Size, n
- Average,
- S. D., σ
- Margin, m
X
![Page 131: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/131.jpg)
Confidence IntervalsE.g. Find:
a) 90% Confidence Interval for
Steps:
- Sample Size, n
- Average,
- S. D., σ
- Margin, m
- CI endpoint, left
X
![Page 132: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/132.jpg)
Confidence IntervalsE.g. Find:
a) 90% Confidence Interval for
Steps:
- Sample Size, n
- Average,
- S. D., σ
- Margin, m
- CI endpoint, left
- CI endpoint, right
X
![Page 133: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/133.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for : [119.6, 128.0]
![Page 134: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/134.jpg)
Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
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Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
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Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Note: same
margin of error
as before
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Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
![Page 138: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/138.jpg)
Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Inputs:
Sample Size
![Page 139: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/139.jpg)
Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Inputs:
Sample Size
S. D.
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Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Inputs:
Sample Size
S. D.
Alpha
![Page 141: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/141.jpg)
Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Careful: parameter α
![Page 142: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/142.jpg)
Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Careful: parameter α is:
2 tailed outer area
![Page 143: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/143.jpg)
Confidence Intervals
An EXCEL shortcut:
CONFIDENCE
Careful: parameter α is:
2 tailed outer area
So for level = 0.90, α = 0.10
![Page 144: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/144.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for μ: [119.6, 128.0]
![Page 145: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/145.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for μ: [119.6, 128.0]
b) 95% CI for μ: [118.7, 128.9]
![Page 146: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/146.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for μ: [119.6, 128.0]
b) 95% CI for μ: [118.7, 128.9]
c) 99% CI for μ: [117.1, 130.5]
![Page 147: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/147.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for μ: [119.6, 128.0]
b) 95% CI for μ: [118.7, 128.9]
c) 99% CI for μ: [117.1, 130.5]
d) How do the CIs change as the confidence level increases?
![Page 148: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/148.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for μ: [119.6, 128.0]
b) 95% CI for μ: [118.7, 128.9]
c) 99% CI for μ: [117.1, 130.5]
d) How do the CIs change as the confidence level increases?
– Intervals get longer
![Page 149: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/149.jpg)
Confidence IntervalsE.g. Find:
a) 90% CI for μ: [119.6, 128.0]
b) 95% CI for μ: [118.7, 128.9]
c) 99% CI for μ: [117.1, 130.5]
d) How do the CIs change as the confidence level increases?
– Intervals get longer– Reflects higher demand for accuracy
![Page 150: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/150.jpg)
Confidence Intervals
HW: 6.11 (use Excel to draw curve &
shade by hand)
6.13, 6.14 (7.30,7.70, wider)
6.16 (n = 2673, so CLT gives Normal)
![Page 151: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/151.jpg)
Choice of Sample Size
Additional use of margin of error idea
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Choice of Sample Size
Additional use of margin of error idea
Background: distributions
Small n Large n
X
n
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Choice of Sample Size
Could choose n to make = desired valuen
![Page 154: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/154.jpg)
Choice of Sample Size
Could choose n to make = desired value
But S. D. is not very interpretable
n
![Page 155: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/155.jpg)
Choice of Sample Size
Could choose n to make = desired value
But S. D. is not very interpretable, so make “margin of error”, m = desired value
n
![Page 156: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/156.jpg)
Choice of Sample Size
Could choose n to make = desired value
But S. D. is not very interpretable, so make “margin of error”, m = desired value
Then get: “ is within m units of ,
95% of the time”
n
X
![Page 157: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/157.jpg)
Choice of Sample Size
Given m, how do we find n?
![Page 158: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/158.jpg)
Choice of Sample Size
Given m, how do we find n?
Solve for n (the equation):
mXP 95.0
![Page 159: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/159.jpg)
Choice of Sample Size
Given m, how do we find n?
Solve for n (the equation):
(where is n in this?)
mXP 95.0
![Page 160: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/160.jpg)
Choice of Sample Size
Given m, how do we find n?
Solve for n (the equation):
(use of “standardization”)
n
mn
XPmXP
95.0
![Page 161: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/161.jpg)
Choice of Sample Size
Given m, how do we find n?
Solve for n (the equation):
n
mn
XPmXP
95.0
nm
ZP
![Page 162: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/162.jpg)
Choice of Sample Size
Given m, how do we find n?
Solve for n (the equation):
[so use NORMINV & Stand. Normal, N(0,1)]
n
mn
XPmXP
95.0
nm
ZP
![Page 163: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/163.jpg)
Choice of Sample Size
Graphically, find m so that:
Area = 0.95
nm
![Page 164: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/164.jpg)
Choice of Sample Size
Graphically, find m so that:
Area = 0.95 Area = 0.975
nm
nm
![Page 165: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/165.jpg)
Choice of Sample Size
Thus solve:
1,0,975.0NORMINVn
m
![Page 166: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/166.jpg)
Choice of Sample Size
Thus solve:
1,0,975.0NORMINVn
m
1,0,975.0NORMINVm
n
![Page 167: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/167.jpg)
Choice of Sample Size
Thus solve:
2
1,0,975.0
NORMINVm
n
1,0,975.0NORMINVn
m
1,0,975.0NORMINVm
n
![Page 168: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/168.jpg)
Choice of Sample Size
(put this on list of formulas)
2
1,0,975.0
NORMINVm
n
![Page 169: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/169.jpg)
Choice of Sample Size
Numerical fine points:
2
1,0,975.0
NORMINVm
n
![Page 170: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/170.jpg)
Choice of Sample Size
Numerical fine points:
• Change this for coverage prob. ≠ 0.95
2
1,0,975.0
NORMINVm
n
![Page 171: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/171.jpg)
Choice of Sample Size
Numerical fine points:
• Change this for coverage prob. ≠ 0.95
• Round decimals upwards,
To be “sure of desired coverage”
2
1,0,975.0
NORMINVm
n
![Page 172: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/172.jpg)
Choice of Sample Size
EXCEL Implementation:
Class Example 11, Part 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg11.xls
2
1,0,975.0
NORMINVm
n
![Page 173: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/173.jpg)
Choice of Sample Size
Class Example 11, Part 2:
Recall:
Corn Yield Data
![Page 174: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/174.jpg)
Choice of Sample Size
Class Example 11, Part 2:
Recall:
Corn Yield Data
Gave X
![Page 175: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/175.jpg)
Choice of Sample Size
Class Example 11, Part 2:
Recall:
Corn Yield Data
Gave
Assumed σ = 10
X
![Page 176: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/176.jpg)
Choice of Sample Size
Class Example 11, Part 2:
Recall:
Corn Yield Data
Resulted in margin of error, m
![Page 177: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/177.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
![Page 178: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/178.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from: 2
1,0,95.0
NORMINVm
n
![Page 179: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/179.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from:
(recall 90% central area,
so use 95% cutoff)
2
1,0,95.0
NORMINVm
n
![Page 180: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/180.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from: 2
1,0,95.0
NORMINVm
n
![Page 181: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/181.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from: 2
1,0,95.0
NORMINVm
n
![Page 182: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/182.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from: 2
1,0,95.0
NORMINVm
n
![Page 183: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/183.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from: 2
1,0,95.0
NORMINVm
n
![Page 184: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/184.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
Compute from:
Round up, to be
safe in statement
2
1,0,95.0
NORMINVm
n
![Page 185: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/185.jpg)
Choice of Sample Size
Class Example 11, Part 2:
Excel Function to round up:
CIELING
![Page 186: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/186.jpg)
Choice of Sample Size
Class Example 11, Part 2:
How large should n be to give smaller (90%) margin of error, say m = 2?
n = 68
![Page 187: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/187.jpg)
Choice of Sample Size
Now ask for higher confidence level:
How large should n be to give smaller (99%) margin of error, say m = 2?
![Page 188: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/188.jpg)
Choice of Sample Size
Now ask for higher confidence level:
How large should n be to give smaller (99%) margin of error, say m = 2?
Similar computations:
n = 166
![Page 189: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/189.jpg)
Choice of Sample Size
Now ask for smaller margin:
How large should n be to give smaller (99%) margin of error, say m = 0.2?
![Page 190: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/190.jpg)
Choice of Sample Size
Now ask for smaller margin:
How large should n be to give smaller (99%) margin of error, say m = 0.2?
Similar computations:
n = 16588
![Page 191: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/191.jpg)
Choice of Sample Size
Now ask for smaller margin:
How large should n be to give smaller (99%) margin of error, say m = 0.2?
Similar computations:
n = 16588
Note: serious
round up
![Page 192: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/192.jpg)
Choice of Sample Size
Now ask for smaller margin:
How large should n be to give smaller (99%) margin of error, say m = 0.2?
Similar computations:
n = 16588
(10 times the accuracy requires
100 times as much data)
![Page 193: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/193.jpg)
Choice of Sample Size
Now ask for smaller margin:
How large should n be to give smaller (99%) margin of error, say m = 0.2?
Similar computations:
n = 16588
(10 times the accuracy requires
100 times as much data)
(Law of Averages: Square Root)
![Page 194: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/194.jpg)
Choice of Sample Size
HW: 6.29, 6.30 (52), 6.31
2
1,0,95.0
NORMINVm
n
![Page 195: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/195.jpg)
And now for somethingcompletely different….
An interesting advertisement:
http://www.albinoblacksheep.com/flash/honda.php
![Page 196: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/196.jpg)
C.I.s for proportionsRecall:
Counts: pnpnppnBiX XX 1,,,~
![Page 197: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/197.jpg)
C.I.s for proportionsRecall:
Counts:
Sample Proportions:
pnpnppnBiX XX 1,,,~
npp
pnX
p pp
1,,ˆ ˆˆ
![Page 198: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/198.jpg)
C.I.s for proportions
Calculate prob’s with BINOMDIST
![Page 199: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/199.jpg)
C.I.s for proportions
Calculate prob’s with BINOMDIST
(but C.I.s need inverse of probs)
![Page 200: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/200.jpg)
C.I.s for proportions
Calculate prob’s with BINOMDIST,
but note no BINOMINV
![Page 201: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/201.jpg)
C.I.s for proportions
Calculate prob’s with BINOMDIST,
but note no BINOMINV,
so instead use Normal Approximation
Recall:
![Page 202: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/202.jpg)
Normal Approx. to BinomialExample: from StatsPortal
http://courses.bfwpub.com/ips6e.php
For Bi(n,p):
Control n
Control p
See Prob. Histo.
Compare to fit
(by mean & sd)
Normal dist’n
![Page 203: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/203.jpg)
C.I.s for proportions
Recall Normal Approximation to Binomial
![Page 204: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/204.jpg)
C.I.s for proportions
Recall Normal Approximation to Binomial:
For 101&10 pnnp
![Page 205: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/205.jpg)
C.I.s for proportions
Recall Normal Approximation to Binomial:
For
is approximatelyX pnpnpN 1,
101&10 pnnp
![Page 206: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/206.jpg)
C.I.s for proportions
Recall Normal Approximation to Binomial:
For
is approximately
is approximately
npp
pN1
,
X pnpnpN 1,
p
101&10 pnnp
![Page 207: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/207.jpg)
C.I.s for proportions
Recall Normal Approximation to Binomial:
For
is approximately
is approximately
So use NORMINV
npp
pN1
,
X pnpnpN 1,
p
101&10 pnnp
![Page 208: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/208.jpg)
C.I.s for proportions
Recall Normal Approximation to Binomial:
For
is approximately
is approximately
So use NORMINV (and often NORMDIST)
npp
pN1
,
X pnpnpN 1,
p
101&10 pnnp
![Page 209: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/209.jpg)
C.I.s for proportions
Main problem: don’t know p
![Page 210: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/210.jpg)
C.I.s for proportions
Main problem: don’t know p
Solution: Depends on context:
CIs or hypothesis tests
![Page 211: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/211.jpg)
C.I.s for proportions
Main problem: don’t know p
Solution: Depends on context:
CIs or hypothesis tests
Different from Normal
![Page 212: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/212.jpg)
C.I.s for proportions
Main problem: don’t know p
Solution: Depends on context:
CIs or hypothesis tests
Different from Normal, since now mean and
sd are linked
![Page 213: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/213.jpg)
C.I.s for proportions
Main problem: don’t know p
Solution: Depends on context:
CIs or hypothesis tests
Different from Normal, since now mean and
sd are linked, with both depending on
p
![Page 214: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/214.jpg)
C.I.s for proportions
Main problem: don’t know p
Solution: Depends on context:
CIs or hypothesis tests
Different from Normal, since now mean and
sd are linked, with both depending on
p, instead of separate μ & σ.
![Page 215: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/215.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
95%
npp
Npp1
,0~ˆ
m m
![Page 216: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/216.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
95% 0.975
npp
Npp1
,0~ˆ
m m m
![Page 217: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/217.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
95% 0.975
So:
npp
Npp1
,0~ˆ
nppNORMINVm /1,0,975.0
m m m
![Page 218: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/218.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
nppNORMINVm /1,0,975.0
![Page 219: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/219.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
Continuing problem: Unknown
nppNORMINVm /1,0,975.0
p
![Page 220: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/220.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
Continuing problem: Unknown
Solution 1: “Best Guess”
nppNORMINVm /1,0,975.0
p
![Page 221: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/221.jpg)
C.I.s for proportions
Case 1: Margin of Error and CIs:
Continuing problem: Unknown
Solution 1: “Best Guess”
Replace by
nppNORMINVm /1,0,975.0
p
p
p
![Page 222: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/222.jpg)
C.I.s for proportionsSolution 2: “Conservative”
![Page 223: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/223.jpg)
C.I.s for proportionsSolution 2: “Conservative”
Idea: make sd (and thus m) as large as possible
![Page 224: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/224.jpg)
C.I.s for proportionsSolution 2: “Conservative”
Idea: make sd (and thus m) as large as possible
(makes no sense for Normal)
![Page 225: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/225.jpg)
C.I.s for proportionsSolution 2: “Conservative”
Idea: make sd (and thus m) as large as possible
(makes no sense for Normal)
pppppf 21
![Page 226: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/226.jpg)
C.I.s for proportionsSolution 2: “Conservative”
Idea: make sd (and thus m) as large as possible
(makes no sense for Normal)
pppppf 21
![Page 227: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/227.jpg)
C.I.s for proportionsSolution 2: “Conservative”
Idea: make sd (and thus m) as large as possible
(makes no sense for Normal)
zeros at 0 & 1
pppppf 21
![Page 228: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/228.jpg)
C.I.s for proportionsSolution 2: “Conservative”
Idea: make sd (and thus m) as large as possible
(makes no sense for Normal)
zeros at 0 & 1
max at 2/1p
pppppf 21
![Page 229: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/229.jpg)
C.I.s for proportions
Solution 1: “Conservative”
Can check by calculus
so 41
21
121
1max]1,0[
pp
p
![Page 230: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/230.jpg)
C.I.s for proportions
Solution 1: “Conservative”
Can check by calculus
so
Thus nNORMINVm /4/1,0,975.0
41
21
121
1max]1,0[
pp
p
![Page 231: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/231.jpg)
C.I.s for proportions
Solution 1: “Conservative”
Can check by calculus
so
Thus nNORMINVm /4/1,0,975.0
41
21
121
1max]1,0[
pp
p
nsqrtNORMINV *2/1,0,975.0
![Page 232: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/232.jpg)
C.I.s for proportions
Example: Old Text Problem 8.8
![Page 233: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/233.jpg)
C.I.s for proportions
Example: Old Text Problem 8.8
Power companies spend time and money trimming trees to keep branches from falling on lines.
![Page 234: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/234.jpg)
C.I.s for proportions
Example: Old Text Problem 8.8
Power companies spend time and money trimming trees to keep branches from falling on lines. Chemical treatment can stunt tree growth, but too much may kill the tree.
![Page 235: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/235.jpg)
C.I.s for proportions
Example: Old Text Problem 8.8
Power companies spend time and money trimming trees to keep branches from falling on lines. Chemical treatment can stunt tree growth, but too much may kill the tree. In an experiment on 216 trees, 41 died.
![Page 236: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/236.jpg)
C.I.s for proportions
Example: Old Text Problem 8.8
Power companies spend time and money trimming trees to keep branches from falling on lines. Chemical treatment can stunt tree growth, but too much may kill the tree. In an experiment on 216 trees, 41 died. Give a 99% CI for the proportion expected to die from this treatment.
![Page 237: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/237.jpg)
C.I.s for proportions
Example: Old Text Problem 8.8
Solution: Class example 12, part 1http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg12.xls
![Page 238: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/238.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
![Page 239: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/239.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
![Page 240: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/240.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
Sample Prop.,
Check Normal
Approximation
p
![Page 241: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/241.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
Sample Prop.,
Check Normal
Approximation
p
![Page 242: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/242.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
Sample Prop.,
Best Guess
Margin of Error
p
![Page 243: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/243.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
Sample Prop.,
Best Guess
Margin of Error
p
![Page 244: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/244.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
Sample Prop.,
Best Guess
Margin of Error
(Recall 99% level
& 2 tails…)
p
![Page 245: Last Time Central Limit Theorem –Illustrations –How large n? –Normal Approximation to Binomial Statistical Inference –Estimate unknown parameters –Unbiasedness](https://reader030.vdocuments.us/reader030/viewer/2022032606/56649eb55503460f94bbdea3/html5/thumbnails/245.jpg)
C.I.s for proportions
Class e.g. 12, part 1
Sample Size, n
Data Count, X
Sample Prop.,
Best Guess
Margin of Error
Conservative
Margin of Error
p
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C.I.s for proportions
Class e.g. 12, part 1
Best Guess CI:
[0.121, 0.259]
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C.I.s for proportions
Class e.g. 12, part 1
Best Guess CI:
[0.121, 0.259]
Conservative CI:
[0.102, 0.277]
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C.I.s for proportionsExample: Old Text Problem 8.8
Solution: Class example 12, part 1http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg12.xls
Note: Conservative is bigger
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C.I.s for proportionsExample: Old Text Problem 8.8
Solution: Class example 12, part 1http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg12.xls
Note: Conservative is bigger
Since 5.019.0ˆ p
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C.I.s for proportionsExample: Old Text Problem 8.8
Solution: Class example 12, part 1http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg12.xls
Note: Conservative is bigger
Since
Big gap
5.019.0ˆ p
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C.I.s for proportionsExample: Old Text Problem 8.8
Solution: Class example 12, part 1http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg12.xls
Note: Conservative is bigger
Since
Big gap
So may pay substantial
price for being “safe”
5.019.0ˆ p
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C.I.s for proportionsHW:
8.7
Do both best-guess and conservative CIs:
8.11, 8.13a, 8.19