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)

Administrative Matters

Midterm II, coming Tuesday, April 6

Administrative Matters

Midterm II, coming Tuesday, April 6

• Numerical answers:– No computers, no calculators

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)

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

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.)

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

Administrative Matters

Midterm II, coming Tuesday, April 6

• Material Covered:

HW 6 – HW 10

Administrative Matters

Midterm II, coming Tuesday, April 6

• Material Covered:

HW 6 – HW 10

– Note: due Thursday, April 2

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)

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

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)

Study Suggestions

1. Work an Old Exama) On Blackboard

b) Course Information Section

Study Suggestions

1. Work an Old Exama) On Blackboard

b) Course Information Section

c) Afterwards, check against given solutions

Study Suggestions

1. Work an Old Exama) On Blackboard

b) Course Information Section

c) Afterwards, check against given solutions

2. Rework HW problems

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

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”)

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

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

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?)

Extreme Case of CLTConsequences:

roughly

roughly

Terminology: Called

The Normal Approximation to the Binomial

p npppN 1,

X pnpnpN 1,

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}

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

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)

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

Statistical InferenceAnother view:

Form conclusions by

Statistical InferenceAnother view:

Form conclusions by

quantifying uncertainty

Statistical InferenceAnother view:

Form conclusions by

quantifying uncertainty

(will study several approaches, first is…)

Confidence Intervals

Background:

Confidence Intervals

Background:

The sample mean, , is an “estimate”

of the population mean,

X

Confidence Intervals

Background:

The sample mean, , is an “estimate”

of the population mean,

How accurate?

X

Confidence Intervals

Background:

The sample mean, , is an “estimate”

of the population mean,

How accurate?

(there is “variability”, how

much?)

X

Confidence IntervalsIdea:

Since a point estimate

(e.g. or )X p

Confidence IntervalsIdea:

Since a point estimate

is never exactly right

(in particular ) 0XP

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)

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)

Confidence IntervalsE.g. ,~,,1 NXX n

Confidence IntervalsE.g. with σ known ,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Think: measurement error

,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Think: measurement error

Each measurement is Normal

,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Think: measurement error

Each measurement is Normal

Known accuracy (maybe)

,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Think: population modeling

,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Think: population modeling

Normal population

,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Think: population modeling

Normal population

Known s.d.

(a stretch, really need to improve)

,~,,1 NXX n

Confidence IntervalsE.g. with σ known

Recall the Sampling Distribution:

,~,,1 NXX n

nNX

,~

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

,~

Confidence IntervalsE.g. with σ known

Recall the Sampling Distribution:

Use to analyze variation

,~,,1 NXX n

nNX

,~

Confidence Intervals

Understand error as:

(normal density quantifies

randomness in )

ndistX '

X

Confidence Intervals

Understand error as:

(distribution centered at μ)

ndistX '

Confidence Intervals

Understand error as:

(spread: s.d. = )

ndistX 'n

n

Confidence Intervals

Understand error as:

How to explain to untrained consumers?

ndistX 'n

Confidence Intervals

Understand error as:

How to explain to untrained consumers?

(who don’t know randomness,

distributions, normal curves)

ndistX 'n

Confidence Intervals

Approach: present an interval

Confidence Intervals

Approach: present an interval

With endpoints:

Estimate +- margin of error

Confidence Intervals

Approach: present an interval

With endpoints:

Estimate +- margin of error

I.e. mX

Confidence Intervals

Approach: present an interval

With endpoints:

Estimate +- margin of error

I.e.

reflecting variability

mX

Confidence Intervals

Approach: present an interval

With endpoints:

Estimate +- margin of error

I.e.

reflecting variability

mX

Confidence Intervals

Approach: present an interval

With endpoints:

Estimate +- margin of error

I.e.

reflecting variability

How to choose ?

mX

m

Confidence Intervals

Choice of Confidence Interval Radius

Confidence Intervals

Choice of Confidence Interval Radius,

i.e. margin of error, m

Confidence Intervals

Choice of Confidence Interval Radius,

i.e. margin of error, :

Notes:

• No Absolute Range (i.e. including “everything”) is available

m

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

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

Confidence IntervalsChoice of margin of error, m

Confidence IntervalsChoice of margin of error, :Approach:• Choose a Confidence Level

m

Confidence IntervalsChoice of margin of error, :Approach:• Choose a Confidence Level• Often 0.95

m

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

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

Confidence Intervals

E.g. For confidence level 0.95, want

0.95 = Area

Confidence Intervals

E.g. For confidence level 0.95, want

distribution

0.95 = Area

X

Confidence Intervals

E.g. For confidence level 0.95, want

distribution

0.95 = Area

= margin of errorm

X

Confidence Intervals

Computation: Recall NORMINV

Confidence Intervals

Computation: Recall NORMINV takes

areas (probs)

Confidence Intervals

Computation: Recall NORMINV takes

areas (probs), and returns cutoffs

Confidence Intervals

Computation: Recall NORMINV takes

areas (probs), and returns cutoffs

Issue: NORMINV works with lower areas

Confidence Intervals

Computation: Recall NORMINV takes

areas (probs), and returns cutoffs

Issue: NORMINV works with lower areas

Note: lower tail

included

Confidence Intervals

So adapt needed probs to lower areas….

Confidence Intervals

So adapt needed probs to lower areas….

When inner area = 0.95,

Confidence Intervals

So adapt needed probs to lower areas….

When inner area = 0.95,

Right tail = 0.025

Confidence Intervals

So adapt needed probs to lower areas….

When inner area = 0.95,

Right tail = 0.025

Shaded Area = 0.975

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

Confidence Intervals

Need to compute:

nNORMINV

,,975.0

Confidence Intervals

Need to compute:

Major problem: is unknown

nNORMINV

,,975.0

Confidence Intervals

Need to compute:

Major problem: is unknown

• But should answer depend on ?

nNORMINV

,,975.0

Confidence Intervals

Need to compute:

Major problem: is unknown

• But should answer depend on ?

• “Accuracy” is only about spread

nNORMINV

,,975.0

Confidence Intervals

Need to compute:

Major problem: is unknown

• But should answer depend on ?

• “Accuracy” is only about spread

• Not centerpoint

nNORMINV

,,975.0

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

Confidence Intervals

Approach to unknown

Confidence Intervals

Approach to unknown :

Recenter, i.e. look at dist’n

X

Confidence Intervals

Approach to unknown :

Recenter, i.e. look at dist’n

X

Confidence Intervals

Approach to unknown :

Recenter, i.e. look at dist’n

Key concept:

Centered at 0

X

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

Confidence Intervals

Computation of:

nNORMINVm

,0,975.0

Confidence Intervals

Computation of:

Smaller Problem: Don’t know

nNORMINVm

,0,975.0

Confidence Intervals

Computation of:

Smaller Problem: Don’t know

Approach 1: Estimate with

(natural approach: use estimate)

nNORMINVm

,0,975.0

s

Confidence Intervals

Computation of:

Smaller Problem: Don’t know

Approach 1: Estimate with

• Leads to complications

nNORMINVm

,0,975.0

s

Confidence Intervals

Computation of:

Smaller Problem: Don’t know

Approach 1: Estimate with

• Leads to complications

• Will study later

nNORMINVm

,0,975.0

s

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

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

~1?

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

~2?

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

~7?

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

~10?

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

Early Approach:

Use data to choose

window width

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

Challenge:

Not enough info in

data for good choice

Research Corner

How many bumps in stamps data?

Kernel Density Estimates

Depends on Window

Alternate Approach:

Scale Space

Research Corner

Scale Space:

Main Idea:

• Don’t try to choose window width

Research Corner

Scale Space:

Main Idea:

• Don’t try to choose window width

• Instead use all of them

Research Corner

Scale Space:

Main Idea:

• Don’t try to choose window width

• Instead use all of them

• Terminology from Computer Vision

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”)

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)

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)

Research Corner

Scale Space:

View 1: Rainbow colored movie

Research Corner

Scale Space:

View 2: Rainbow

colored overlay

Research Corner

Scale Space:

View 3: Rainbow

colored surface

Research Corner

Scale Space:

Main Idea:

• Don’t try to choose window width

• Instead use all of them

Challenge: how to do statistical inference?

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?

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)

Research Corner

Scale Space:

Challenge: how to do statistical inference?

Which bumps are really there?

(i.e. statistically significant)

Research Corner

Scale Space:

Challenge: how to do statistical inference?

Which bumps are really there?

(i.e. statistically significant)

Address this next time

Confidence Intervals

E.g. Crop researchers plant 15 plots

with a new variety of corn.

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

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

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

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

Confidence IntervalsE.g. Find:

a) 90% Confidence

Interval for

Use Excel

Confidence IntervalsE.g. Find:

a) 90% Confidence

Interval for

Use Excel

Data in C8:C22

Confidence IntervalsE.g. Find:

a) 90% Confidence Interval for

Steps:

- Sample Size, n

Confidence IntervalsE.g. Find:

a) 90% Confidence Interval for

Steps:

- Sample Size, n

- Average,

X

Confidence IntervalsE.g. Find:

a) 90% Confidence Interval for

Steps:

- Sample Size, n

- Average,

- S. D., σ

X

Confidence IntervalsE.g. Find:

a) 90% Confidence Interval for

Steps:

- Sample Size, n

- Average,

- S. D., σ

- Margin, m

X

Confidence IntervalsE.g. Find:

a) 90% Confidence Interval for

Steps:

- Sample Size, n

- Average,

- S. D., σ

- Margin, m

- CI endpoint, left

X

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

Confidence IntervalsE.g. Find:

a) 90% CI for : [119.6, 128.0]

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Note: same

margin of error

as before

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Inputs:

Sample Size

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Inputs:

Sample Size

S. D.

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Inputs:

Sample Size

S. D.

Alpha

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Careful: parameter α

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Careful: parameter α is:

2 tailed outer area

Confidence Intervals

An EXCEL shortcut:

CONFIDENCE

Careful: parameter α is:

2 tailed outer area

So for level = 0.90, α = 0.10

Confidence IntervalsE.g. Find:

a) 90% CI for μ: [119.6, 128.0]

Confidence IntervalsE.g. Find:

a) 90% CI for μ: [119.6, 128.0]

b) 95% CI for μ: [118.7, 128.9]

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]

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?

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

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

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)

Choice of Sample Size

Additional use of margin of error idea

Choice of Sample Size

Additional use of margin of error idea

Background: distributions

Small n Large n

X

n

Choice of Sample Size

Could choose n to make = desired valuen

Choice of Sample Size

Could choose n to make = desired value

But S. D. is not very interpretable

n

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

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

Choice of Sample Size

Given m, how do we find n?

Choice of Sample Size

Given m, how do we find n?

Solve for n (the equation):

mXP 95.0

Choice of Sample Size

Given m, how do we find n?

Solve for n (the equation):

(where is n in this?)

mXP 95.0

Choice of Sample Size

Given m, how do we find n?

Solve for n (the equation):

(use of “standardization”)

n

mn

XPmXP

95.0

Choice of Sample Size

Given m, how do we find n?

Solve for n (the equation):

n

mn

XPmXP

95.0

nm

ZP

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

Choice of Sample Size

Graphically, find m so that:

Area = 0.95

nm

Choice of Sample Size

Graphically, find m so that:

Area = 0.95 Area = 0.975

nm

nm

Choice of Sample Size

Thus solve:

1,0,975.0NORMINVn

m

Choice of Sample Size

Thus solve:

1,0,975.0NORMINVn

m

1,0,975.0NORMINVm

n

Choice of Sample Size

Thus solve:

2

1,0,975.0

NORMINVm

n

1,0,975.0NORMINVn

m

1,0,975.0NORMINVm

n

Choice of Sample Size

(put this on list of formulas)

2

1,0,975.0

NORMINVm

n

Choice of Sample Size

Numerical fine points:

2

1,0,975.0

NORMINVm

n

Choice of Sample Size

Numerical fine points:

• Change this for coverage prob. ≠ 0.95

2

1,0,975.0

NORMINVm

n

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

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

Choice of Sample Size

Class Example 11, Part 2:

Recall:

Corn Yield Data

Choice of Sample Size

Class Example 11, Part 2:

Recall:

Corn Yield Data

Gave X

Choice of Sample Size

Class Example 11, Part 2:

Recall:

Corn Yield Data

Gave

Assumed σ = 10

X

Choice of Sample Size

Class Example 11, Part 2:

Recall:

Corn Yield Data

Resulted in margin of error, m

Choice of Sample Size

Class Example 11, Part 2:

How large should n be to give smaller (90%) margin of error, say m = 2?

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

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

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

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

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

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

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

Choice of Sample Size

Class Example 11, Part 2:

Excel Function to round up:

CIELING

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

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?

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

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?

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

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

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)

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)

Choice of Sample Size

HW: 6.29, 6.30 (52), 6.31

2

1,0,95.0

NORMINVm

n

And now for somethingcompletely different….

An interesting advertisement:

http://www.albinoblacksheep.com/flash/honda.php

C.I.s for proportionsRecall:

Counts: pnpnppnBiX XX 1,,,~

C.I.s for proportionsRecall:

Counts:

Sample Proportions:

pnpnppnBiX XX 1,,,~

npp

pnX

p pp

1,,ˆ ˆˆ

C.I.s for proportions

Calculate prob’s with BINOMDIST

C.I.s for proportions

Calculate prob’s with BINOMDIST

(but C.I.s need inverse of probs)

C.I.s for proportions

Calculate prob’s with BINOMDIST,

but note no BINOMINV

C.I.s for proportions

Calculate prob’s with BINOMDIST,

but note no BINOMINV,

so instead use Normal Approximation

Recall:

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

C.I.s for proportions

Recall Normal Approximation to Binomial

C.I.s for proportions

Recall Normal Approximation to Binomial:

For 101&10 pnnp

C.I.s for proportions

Recall Normal Approximation to Binomial:

For

is approximatelyX pnpnpN 1,

101&10 pnnp

C.I.s for proportions

Recall Normal Approximation to Binomial:

For

is approximately

is approximately

npp

pN1

,

X pnpnpN 1,

p

101&10 pnnp

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

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

C.I.s for proportions

Main problem: don’t know p

C.I.s for proportions

Main problem: don’t know p

Solution: Depends on context:

CIs or hypothesis tests

C.I.s for proportions

Main problem: don’t know p

Solution: Depends on context:

CIs or hypothesis tests

Different from Normal

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

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

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.

C.I.s for proportions

Case 1: Margin of Error and CIs:

95%

npp

Npp1

,0~ˆ

m m

C.I.s for proportions

Case 1: Margin of Error and CIs:

95% 0.975

npp

Npp1

,0~ˆ

m m m

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

C.I.s for proportions

Case 1: Margin of Error and CIs:

nppNORMINVm /1,0,975.0

C.I.s for proportions

Case 1: Margin of Error and CIs:

Continuing problem: Unknown

nppNORMINVm /1,0,975.0

p

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

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

C.I.s for proportionsSolution 2: “Conservative”

C.I.s for proportionsSolution 2: “Conservative”

Idea: make sd (and thus m) as large as possible

C.I.s for proportionsSolution 2: “Conservative”

Idea: make sd (and thus m) as large as possible

(makes no sense for Normal)

C.I.s for proportionsSolution 2: “Conservative”

Idea: make sd (and thus m) as large as possible

(makes no sense for Normal)

pppppf 21

C.I.s for proportionsSolution 2: “Conservative”

Idea: make sd (and thus m) as large as possible

(makes no sense for Normal)

pppppf 21

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

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

C.I.s for proportions

Solution 1: “Conservative”

Can check by calculus

so 41

21

121

1max]1,0[

pp

p

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

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

C.I.s for proportions

Example: Old Text Problem 8.8

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.

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.

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.

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.

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

C.I.s for proportions

Class e.g. 12, part 1

Sample Size, n

C.I.s for proportions

Class e.g. 12, part 1

Sample Size, n

Data Count, X

C.I.s for proportions

Class e.g. 12, part 1

Sample Size, n

Data Count, X

Sample Prop.,

Check Normal

Approximation

p

C.I.s for proportions

Class e.g. 12, part 1

Sample Size, n

Data Count, X

Sample Prop.,

Check Normal

Approximation

p

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

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

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

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

C.I.s for proportions

Class e.g. 12, part 1

Best Guess CI:

[0.121, 0.259]

C.I.s for proportions

Class e.g. 12, part 1

Best Guess CI:

[0.121, 0.259]

Conservative CI:

[0.102, 0.277]

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

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

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

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

C.I.s for proportionsHW:

8.7

Do both best-guess and conservative CIs:

8.11, 8.13a, 8.19