Transcript plug in: get out

Administrative Matters
Midterm II Results
Take max of two midterm scores:
Administrative Matters
Midterm II Results
Take max of two midterm scores
Approx grades:
92-100 A
82-92
B
70-82
C
60-70
D
0 – 60
F
Last Time
• Confidence Intervals
– For proportions (Binomial)
• Choice of sample size
– For Normal Mean
– For proportions (Binomial)
• Interpretation of Confidence Intervals
Reading In Textbook
Approximate Reading for Today’s Material:
Pages 493-501, 422-435, 447-467
Approximate Reading for Next Class:
Pages 422-435, 372-390
Sample Size for Proportions
i.e. find n so that:
Now solve to get:
m
 NORMINV (0.975,0,1)
p 1  p 
n
NORMINV 0.975,0,1 p1  p 
n
m
NORMINV 0.975,0,1 

n
 p1  p 
m


2
(good candidate for list of formulas)
Sample Size for Proportions
i.e. find n so that:
Now solve to get:
m
 NORMINV (0.975,0,1)
p 1  p 
n
NORMINV 0.975,0,1 p1  p 
n
m
NORMINV 0.975,0,1 

n
 p1  p 
m


2
Problem:
don’t know p
Sample Size for Proportions
Solution 1:
Use
Best Guess
p̂ from:
–
Earlier Study
–
Previous Experience
–
Prior Idea
Sample Size for Proportions
Solution 2:
Recall
Conservative
max p1  p   1
p 0,1
4
So “safe” to use:
NORMINV 0.975,0,1  1

n

m

 4
2
Interpretation of Conf. Intervals
Mathematically:
0.95  P  m  X    m  P X    m
pic 1
pic 2
 PX  m    X  m
3rd interpretation
 Pthe C. I . X  m, X  m " brackets "
Interpretation of Conf. Intervals
Frequentist View:
If repeat the
experiment many times
Interpretation of Conf. Intervals
Frequentist View:
If repeat the
experiment many times,
About 95% of the time, CI’s will contain μ
Interpretation of Conf. Intervals
Frequentist View:
If repeat the
experiment many times,
About 95% of the time, CI’s will contain μ
(and 5% of the time it won’t)
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Shows proper interpretation
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Shows proper interpretation:
–
If repeat drawing the sample
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Shows proper interpretation:
–
If repeat drawing the sample
–
Interval will cover truth 95% of time
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Lower Confidence Level (95%  80%)
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Lower Confidence Level (95%  80%):
–
Shorter confidence intervals
Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
•
Statistical Applets
•
Confidence Intervals
Lower Confidence Level (95%  80%):
–
Shorter confidence intervals
–
Leads to lower hit rate
Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Revisit Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Interpretation of Conf. Intervals
Estimate % of Male Students at UNC
Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Recall:
Q1: Sample of 25 from Class
Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Recall:
Q1: Sample of 25 from Class
Q2: Sample of 25 from any doorway
Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Recall:
Q1: Sample of 25 from Class
Q2: Sample of 25 from any doorway
Q3: Sample of 25 think of names
Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Recall:
Q1: Sample of 25 from Class
Q2: Sample of 25 from any doorway
Q3: Sample of 25 think of names
Q4: Random sample (from phone book)
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class
Compare Q1 and Binomial(25,p)
25
20
15
Q1
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class:
Compare Q1 and Binomial(25,p)
20
15
Q1
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0
0.08
theoretical
25
0
- Compare to
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class:
Compare Q1 and Binomial(25,p)
15
Q1
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0
0.16
- Some bias
20
0.08
theoretical
25
0
- Compare to
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class:
Compare Q1 and Binomial(25,p)
- Compare to
theoretical
- Some bias
25
20
15
Q1
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
- less variation
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q2:
From Doorways
Compare Q2 and Binomial(25,p)
25
20
15
Q2
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q2:
From Doorways:
Compare Q2 and Binomial(25,p)
25
20
15
Q2
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
- No bias
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q2:
From Doorways:
Compare Q2 and Binomial(25,p)
20
15
Q2
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0
0.08
- More variation
25
0
- No bias
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q3:
Think up names
Compare Q3 and Binomial(25,p)
25
20
15
Q3
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q3:
Think up names:
Compare Q3 and Binomial(25,p)
25
20
15
Q3
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
- Upwards bias
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q3:
Think up names:
Compare Q3 and Binomial(25,p)
20
15
Q3
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0
0.08
- More variation
25
0
- Upwards bias
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample
Compare Q4 and Binomial(25,p)
25
20
15
Q4
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample:
Compare Q4 and Binomial(25,p)
25
20
15
Q4
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
0
- Looks better?
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample:
Compare Q4 and Binomial(25,p)
15
Q4
Binomial(25,p)
10
5
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0
0.16
variation?
20
0.08
- Reasonable
25
0
- Looks better?
Interpretation of Conf. Intervals
Histogram analysis: Class Example 7
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample:
Compare Q4 and Binomial(25,p)
- Looks better?
- Reasonable
variation?
25
20
15
Q4
Binomial(25,p)
10
0.96
0.88
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0
0.08
CIs etc.
5
0
- Really need
Interpretation of Conf. Intervals
Now consider C.I. View:
Class Example 13
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls
Interpretation of Conf. Intervals
Now consider C.I. View:
Class Example 13
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls
Explore idea:
CI should cover 90% of time
Interpretation of Conf. Intervals
Class Example 13
Interpretation of Conf. Intervals
Class Example 13
Interpretation of Conf. Intervals
Class Example 13
Interpretation of Conf. Intervals
Class Example 13
Interpretation of Conf. Intervals
Class Example 13
HW # 1:
Q1
0.9
0.8
proportion of males
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
8
15
22
29
36
43
50
57
Student
64
71
78
85
92
99
Interpretation of Conf. Intervals
Class Example 13
Q1: Summarize Coverage
Interpretation of Conf. Intervals
Class Example 13
Q1: Summarize Coverage
94% > 90%
(since sd too small)
Interpretation of Conf. Intervals
Class Example 13
Q2: Summarize Coverage
HW # 1: Q2
1.2
1
(since too
variable)
proportion of males
77% < 90%
0.8
0.6
0.4
0.2
0
1
8
15
22
29
36
43
50
57
-0.2
student
64
71
78
85
92
99
Interpretation of Conf. Intervals
Class Example 13
Q3: Summarize Coverage
HW # 1: Q 3
1.2
(since too
biased)
proportion of males
77% < 90%
1
0.8
0.6
0.4
0.2
0
1
8
15
22
29
36
43
50
57
student
64
71
78
85
92
99
Interpretation of Conf. Intervals
Class Example 13
Q4: Summarize Coverage
HW # 1: Q4
1.2
1
(seems OK?)
proportion of males
87% ≈ 90%
0.8
0.6
0.4
0.2
0
1
8
15
22
29
36
43
50
57
-0.2
student
64
71
78
85
92
99
Interpretation of Conf. Intervals
Class Example 13
Simulate from
Simulated Binomial(25,0.43)
0.9
Binomial
0.8
87% ≈ 90%
(shows within
expected
range)
proportion of males
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
8
15
22
29
36
43
50
57
student
64
71
78
85
92
99
Interpretation of Conf. Intervals
Class Example 13:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls
Q1: SD too small  Too many cover
Q2: SD too big  Too few cover
Q3: Big Bias  Too few cover
Q4: Good sampling  About right
Q5: Simulated Bi  Shows “natural var’n”
Interpretation of Conf. Intervals
HW:
6.20 ($1260, $1540), 6.21
6.28 (but use Excel & make histogram)
Research Corner
Another SiZer analysis:
British Incomes Data
Research Corner
Another SiZer analysis:
British Incomes Data
o
Annual Survey (1985)
Research Corner
Another SiZer analysis:
British Incomes Data
o
Annual Survey (1985)
o
Done in Great Britain
Research Corner
Another SiZer analysis:
British Incomes Data
o
Annual Survey (1985)
o
Done in Great Britain
o
Variable of Interest:
Family Income
Research Corner
Another SiZer analysis:
British Incomes Data
o
Annual Survey (1985)
o
Done in Great Britain
o
Variable of Interest:
Family Income
o
Distribution?
Research Corner
British Incomes Data
SiZer Results:
 1 bump at coarse scale
(expected)
Research Corner
British Incomes Data
SiZer Results:
 1 bump at coarse scale
 2 bumps at medium scale
Research Corner
British Incomes Data
SiZer Results:
 1 bump at coarse scale
 2 bumps at medium scale
(Quite a radical
statement)
Research Corner
British Incomes Data
SiZer Results:
 1 bump at coarse scale
 2 bumps at medium scale
 Finer scale bumps not
statistically significant
Research Corner
British Incomes Data
 2 bumps at medium scale
 Usual models for Incomes
(one bump only)
Research Corner
British Incomes Data
 2 bumps at medium scale
 Usual models for Incomes
(one bump only)
 2 bumps were verified
Research Corner
British Incomes Data
 2 bumps at medium scale
 Usual models for Incomes
(one bump only)
 2 bumps were verified
(in PhD dissertation)
Research Corner
British Incomes Data
 2 bumps at medium scale
 Usual models for Incomes
(one bump only)
 2 bumps were verified
(in PhD dissertation)
 But when worth looking?
Next time
• Add multiple year
plots as well
In:
IncomesAllKDE.mpg
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue:
In sampling distribution
X   ~ N 0, / n 
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue:
In sampling distribution
X   ~ N 0, / n 
Usually σ is unknown
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue:
In sampling distribution
X   ~ N 0, / n 
Usually σ is unknown, so replace with an
estimate, s.
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue:
In sampling distribution
X   ~ N 0, / n 
Usually σ is unknown, so replace with an
estimate, s.
For n large, should be “OK”
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue:
In sampling distribution
X   ~ N 0, / n 
Usually σ is unknown, so replace with an
estimate, s.
For n large, should be “OK”, but what about:
•
n small?
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue:
In sampling distribution
X   ~ N 0, / n 
Usually σ is unknown, so replace with an
estimate, s.
For n large, should be “OK”, but what about:
•
n small?
•
How large is n “large”?
Unknown SD
Goal:
Account for “extra variability in the
s ≈ σ
approximation”
Unknown SD
Goal:
Account for “extra variability in the
s ≈ σ
Mathematics:
approximation”
Assume individual X i ~ N  , 
Unknown SD
Goal:
Account for “extra variability in the
s ≈ σ
Mathematics:
approximation”
Assume individual X i ~ N  , 
I.e.
•
Data have mound shaped histogram
Unknown SD
Goal:
Account for “extra variability in the
s ≈ σ
Mathematics:
approximation”
Assume individual X i ~ N  , 
I.e.
•
Data have mound shaped histogram
•
Recall averages generally normal
Unknown SD
Goal:
Account for “extra variability in the
s ≈ σ
Mathematics:
approximation”
Assume individual X i ~ N  , 
I.e.
•
Data have mound shaped histogram
•
Recall averages generally normal
•
But now must focus on individuals
Unknown SD
Then

X ~ N  , / n

Unknown SD
Then

X ~ N  , / n

X 
So can write:

n
~ N 0,1
Unknown SD
Then

X ~ N  , / n

X 
So can write:

~ N 0,1
n
(recall: standardization (Z-score) idea)
Unknown SD
Then

X ~ N  , / n

X 
So can write:

~ N 0,1
n
(recall: standardization (Z-score) idea,
used in an important way here)
Unknown SD
Then

X ~ N  , / n

X 
So can write:
Replace


n
~ N 0,1
Unknown SD
Then

X ~ N  , / n

X 

So can write:
Replace

by
s
n
~ N 0,1
Unknown SD
Then

X ~ N  , / n

X 

So can write:
Replace

by
~ N 0,1
n
s , then
X 
s
n
Unknown SD
Then

X ~ N  , / n

X 

So can write:
Replace

by
~ N 0,1
n
s , then
has a distribution named
X 
s
n
Unknown SD
Then

X ~ N  , / n

X 

So can write:
Replace

by
~ N 0,1
n
s , then
has a distribution named:
X 
s
n
“t-distribution with n-1 degrees of freedom”
t - Distribution
Notes:
1.
n is a parameter
t - Distribution
Notes:
1.
n is a parameter (like
 ,  , p , )
t - Distribution
Notes:
1.
n is a parameter (like
 ,  , p , )
(Recall: these index families
of probability distributions)
t - Distribution
Notes:
1.
n is a parameter (like  ,  , p, )
that controls “added variability that
comes from the s ≈ σ approximation”
t - Distribution
Notes:
1.
n is a parameter (like  ,  , p, )
that controls “added variability that
comes from the s ≈ σ approximation”
View:
Study Densities,
over degrees of freedom…
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/EgTDist.mpg
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread:
- Lower Peak
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread:
- Lower Peak
- Fatter Tails
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
smaller 5%-tile
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
smaller 5%-tile
larger 99%-tile
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
Makes sense,
since s ≈ σ 
more variation
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 3
All effects are
magnified
Since s ≈ σ approx
gets worse
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 1
Extreme Case
Have terrible
s ≈ σ approx
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
Now try larger d.f.
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 14
All approximations
are better
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 25
Even better
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 25
Even better
- Densities almost
on top
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 25
Even better
- Densities almost
on top
- Quantiles very
close
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 100
Hard to see any
difference
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 100
Hard to see any
difference
Since excellent
s ≈ σ approx
t - Distribution
Notes:
2.
Careful:
set “degrees of freedom” =
= n–1
t - Distribution
Notes:
2.
Careful:
set “degrees of freedom” =
= n–1
(not n)
t - Distribution
Notes:
2.
Careful:
set “degrees of freedom” =
= n–1
•
(not n)
Easy to forget later
t - Distribution
Notes:
2.
Careful:
set “degrees of freedom” =
= n–1
(not n)
•
Easy to forget later
•
Good to add to sheet of notes for exam
t - Distribution
Notes:
3.
Must work with standardized version of
X
t - Distribution
Notes:
3.
Must work with standardized version of
X i.e. X  
s
n
t - Distribution
Notes:
3.
Must work with standardized version of
X i.e. X  
s
n
(will affect how we compute probs….)
t - Distribution
Notes:
3.
•
Must work with standardized version of
X i.e. X  
s
n
No longer can plug mean and SD
into EXCEL formulas
t - Distribution
Notes:
3.
•
Must work with standardized version of
X i.e. X  
s
n
No longer can plug mean and SD
into EXCEL formulas
•
In text standardization was already done
t - Distribution
Notes:
3.
•
Must work with standardized version of
X i.e. X  
s
n
No longer can plug mean and SD
into EXCEL formulas
•
In text standardization was already done,
since used in Normal table calc’ns
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST & TINV
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST & TINV
Caution: these are set up differently from
NORMDIST & NORMINV
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST & TINV
Caution: these are set up differently from
NORMDIST & NORMINV
See Class Example 14
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg14.xls
t - Distribution
Class Example 14:
t - Distribution
Class Example 14:
Calculate Upper Prob
t - Distribution
Class Example 14:
Calculate Upper Prob
t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
(Check TDIST menu)
t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
- cutoff
t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
- cutoff
- d. f.
t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
- cutoff
- d. f.
- upper prob. only
t - Distribution
Class Example 14:
Careful: opposite from NORMDIST
Using TDIST
- cutoff
- d. f.
- upper prob. only
t - Distribution
Class Example 14:
Careful: opposite from NORMDIST
use upper
Using TDIST
- cutoff
- d. f.
- upper prob. only
t - Distribution
Class Example 14:
Careful: opposite from NORMDIST
use upper, NOT lower probs
Using TDIST
- cutoff
- d. f.
- upper prob. only
t - Distribution
Class Example 14:
To compute lower prob
t - Distribution
Class Example 14:
To compute lower prob
Use “1 – trick”, i.e. Not Rule of probability
t - Distribution
Class Example 14:
How about upper prob of negative?
t - Distribution
Class Example 14:
How about upper prob of negative?
Give it a try
t - Distribution
Class Example 14:
How about upper prob of negative?
Give it a try
Get an error message in response
t - Distribution
Class Example 14:
How about upper prob of negative?
Give it a try
Get an error message in response
(Click this for sometimes useful info)
t - Distribution
Class Example 14:
Reason:
TDIST tuned for 2-tailed
t - Distribution
Class Example 14:
Reason:
TDIST tuned for 2-tailed
(where need cutoff > 0)
t - Distribution
Class Example 14:
Reason:
TDIST tuned for 2-tailed
(where need cutoff > 0)
(correct version for CIs and H. tests)
t - Distribution
Class Example 14:
Approach:
t - Distribution
Class Example 14:
Approach:
Use “1 – trick”
t - Distribution
Class Example 14:
Approach:
Use “1 – trick”
(to write as prob. can compute)
t - Distribution
Class Example 14:
For Two-Tailed Prob
t - Distribution
Class Example 14:
For Two-Tailed Prob
TDIST is very convenient
t - Distribution
Class Example 14:
For Two-Tailed Prob
TDIST is very convenient
(much better than NORMDIST)
t - Distribution
Class Example 14:
For Interior Prob
t - Distribution
Class Example 14:
For Interior Prob
Use “1 – trick”
t - Distribution
Class Example 14:
For Interior Prob
Use “1 – trick”
TDIST again very convenient
t - Distribution
Class Example 14:
For Interior Prob
Use “1 – trick”
TDIST again very convenient
(again better than NORMDIST)
t - Distribution
Class Example 14:
Now try increasing d.f.
t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
But converges for larger n
t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
But converges for larger n
To Normal(0,1)
t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
But converges for larger n
To Normal(0,1)
(as expected)
t - Distribution
HW: C23
For T ~ t, with degrees of freedom:
(a) 3 (b) 12 (c) 150 (d) N(0,1)
Find:
i. P{T> 1.7} (0.094, 0.057, 0.046, 0.045)
ii. P{T < 2.14} (0.939, 0.973, 0.983, 0.984)
iii. P{T < -0.74} (0.256, 0.237, 0.230, 0.230)
iv. P{T > -1.83} (0.918, 0.954, 0.965, 0.966)
t - Distribution
HW: C23
v. P{|T| > 1.18} (0.323, 0.261, 0.240, 0.238)
vi. P{|T| < 2.39} (0.903, 0.966, 0.982, 0.983)
vii. P{|T| < -2.74} (0, 0, 0, 0)
And now for something
completely different
“Thinking Outside the Box”
Also Called:
“Lateral Thinking”
And now for something
completely different
Find the word or simple phrase suggested:
death ..... life
And now for something
completely different
Find the word or simple phrase suggested:
death ..... life
life after death
And now for something
completely different
Find the word or simple phrase suggested:
ecnalg
And now for something
completely different
Find the word or simple phrase suggested:
ecnalg
backward glance
And now for something
completely different
Find the word or simple phrase suggested:
He's X himself
And now for something
completely different
Find the word or simple phrase suggested:
He's X himself
He's by himself
And now for something
completely different
Find the word or simple phrase suggested:
THINK
And now for something
completely different
Find the word or simple phrase suggested:
THINK
think big ! !
And now for something
completely different
Find the word or simple phrase suggested:
ababaaabbbbaaaabbbb ababaabbaaabbbb. .
And now for something
completely different
Find the word or simple phrase suggested:
ababaaabbbbaaaabbbb ababaabbaaabbbb. .
long time no 'C'
t - Distribution
Class Example 14:
Next explore TINV
(Inverse function)
t - Distribution
Class Example 14:
Next explore TINV
(Inverse function)
(Given cutoff, find area)
t - Distribution
Class Example 14:
Next explore TINV
t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area)
t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area) & d.f.
t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area) & d.f., find cutoff
t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area) & d.f., find cutoff
(next think carefully about interpretation)
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Now invert this,
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Now invert this,
i.e. given prob.
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Now invert this,
i.e. given prob., find cutoff
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
For same d.f.
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
For same d.f., use resulting prob. as input
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
For same d.f., use resulting prob. as input
But new answer is different
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding?
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding? Try exact value
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding? Try exact value
t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding? Try exact value
Still get wrong answer
t - Distribution
Class Example 14:
Next explore TINV
Reason for inconsistency:
t - Distribution
Class Example 14:
Next explore TINV
Reason for inconsistency:
Works via 2-tailed
t - Distribution
Class Example 14:
Next explore TINV
Reason for inconsistency:
Works via 2-tailed, not 1-tailed, probability
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
Get:
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
Get:
plug in above output
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
Get:
plug in above output, to return to input
EXCEL Functions
Summary:
Normal:
EXCEL Functions
Summary:
Normal:
plug in:
get out:
EXCEL Functions
Summary:
Normal:
NORMDIST:
plug in:
cutoff
get out:
EXCEL Functions
Summary:
Normal:
NORMDIST:
plug in:
cutoff
get out:
area
EXCEL Functions
Summary:
Normal:
NORMDIST:
NORMINV:
plug in:
cutoff
area
get out:
area
EXCEL Functions
Summary:
Normal:
NORMDIST:
NORMINV:
plug in:
cutoff
area
get out:
area
cutoff
EXCEL Functions
Summary:
Normal:
plug in:
get out:
NORMDIST:
cutoff
area
NORMINV:
area
cutoff
(but TDIST is set up really differently)
EXCEL Functions
t distribution:
1 tail:
EXCEL Functions
t distribution:
1 tail:
plug in:
get out:
EXCEL Functions
t distribution:
1 tail:
TDIST:
plug in:
cutoff
get out:
EXCEL Functions
t distribution:
1 tail:
TDIST:
plug in:
cutoff
get out:
area
EXCEL Functions
t distribution:
1 tail:
TDIST:
EXCEL notes:
plug in:
cutoff
get out:
area
- no explicit inverse
EXCEL Functions
t distribution:
1 tail:
TDIST:
EXCEL notes:
plug in:
cutoff
get out:
area
- no explicit inverse
- backwards from Normal…
EXCEL Functions
t distribution:
2 tail:
EXCEL Functions
t distribution:
2 tail:
plug in:
get out:
EXCEL Functions
t distribution:
2 tail:
TDIST:
plug in:
cutoff
get out:
EXCEL Functions
t distribution:
Area
2 tail:
TDIST:
plug in:
cutoff
get out:
area
EXCEL Functions
t distribution:
Area
2 tail:
TDIST:
TINV:
plug in:
cutoff
area
get out:
area
EXCEL Functions
t distribution:
Area
2 tail:
TDIST:
TINV:
plug in:
cutoff
area
get out:
area
cutoff
EXCEL Functions
t distribution:
Area
2 tail:
plug in:
get out:
TDIST:
cutoff
area
TINV:
area
cutoff
(EXCEL note: this one has the inverse)
EXCEL Functions
Note: when need to invert the 1-tail TDIST,
Use twice the area.
EXCEL Functions
Note: when need to invert the 1-tail TDIST,
Use twice the area.
Area = A
EXCEL Functions
Note: when need to invert the 1-tail TDIST,
Use twice the area.
Area = A
Area = 2 A
t - Distribution
HW: C23 (cont.)
viii. C so that 0.05 = P{|T| > C}
(3.18, 2.17, 1.98, 1.96)
ix. C so that 0.99 = P{|T| < C}
(5.84, 3.05, 2.61, 2.58)