Administrative Matters Midterm II Results Take max of two midterm scores:
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Transcript of Administrative Matters Midterm II Results Take max of two midterm scores:
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Administrative Matters
Midterm II Results
Take max of two midterm scores:
![Page 2: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/2.jpg)
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
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Last Time
• Confidence Intervals– For proportions (Binomial)
• Choice of sample size– For Normal Mean– For proportions (Binomial)
• Interpretation of Confidence Intervals
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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
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Sample Size for Proportions
i.e. find so that:
Now solve to get:
(good candidate for list of formulas)
n )1,0,975.0(1
NORMINV
npp
m
m
ppNORMINVn
11,0,975.0
ppm
NORMINVn
1
1,0,975.02
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Sample Size for Proportions
i.e. find so that:
Now solve to get:
Problem: don’t know
n )1,0,975.0(1
NORMINV
npp
m
m
ppNORMINVn
11,0,975.0
p
ppm
NORMINVn
1
1,0,975.02
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Sample Size for Proportions
Solution 1: Best Guess
Use from:
– Earlier Study
– Previous Experience
– Prior Idea
p̂
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Sample Size for Proportions
Solution 2: Conservative
Recall
So “safe” to use:
4
11max1,0
ppp
411,0,975.0
2
mNORMINV
n
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Interpretation of Conf. Intervals
Mathematically:
pic 1 pic 2
3rd interpretation
"",.. bracketsmXmXICtheP
mXPmXmP 95.0
mXmXP
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Interpretation of Conf. Intervals
Frequentist View: If repeat the
experiment many times
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Interpretation of Conf. Intervals
Frequentist View: If repeat the
experiment many times,
About 95% of the time, CI’s will contain μ
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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)
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Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
• Statistical Applets
• Confidence Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
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Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
• Statistical Applets
• Confidence Intervals
Shows proper interpretation
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Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
• Statistical Applets
• Confidence Intervals
Shows proper interpretation:– If repeat drawing the sample
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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
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Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
• Statistical Applets
• Confidence Intervals
Lower Confidence Level (95% 80%)
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Interpretation of Conf. Intervals
Nice Illustration:
Publisher’s Website
• Statistical Applets
• Confidence Intervals
Lower Confidence Level (95% 80%):– Shorter confidence intervals
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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
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Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
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Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Revisit Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
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Interpretation of Conf. Intervals
Estimate % of Male Students at UNC
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Interpretation of Conf. Intervals
Recall Class HW:
Estimate % of Male Students at UNC
Recall:
Q1: Sample of 25 from Class
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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
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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
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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)
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class:
- Compare to
theoretical
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class:
- Compare to
theoretical
- Some bias
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q1:
Sample from Class:
- Compare to
theoretical
- Some bias
- less variation
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q2:
From Doorways
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q2:
From Doorways:
- No bias
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q2:
From Doorways:
- No bias
- More variation
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q3:
Think up names
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q3:
Think up names:
- Upwards bias
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q3:
Think up names:
- Upwards bias
- More variation
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample:
- Looks better?
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample:
- Looks better?
- Reasonable
variation?
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Interpretation of Conf. Intervals
Histogram analysis: Class Example 7http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls
Q4:
Random Sample:
- Looks better?
- Reasonable
variation?
- Really need
CIs etc.
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Interpretation of Conf. Intervals
Now consider C.I. View:
Class Example 13http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls
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Interpretation of Conf. Intervals
Now consider C.I. View:
Class Example 13http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls
Explore idea:
CI should cover 90% of time
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Interpretation of Conf. Intervals
Class Example 13
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Interpretation of Conf. Intervals
Class Example 13
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Interpretation of Conf. Intervals
Class Example 13
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Interpretation of Conf. Intervals
Class Example 13
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Interpretation of Conf. Intervals
Class Example 13
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Interpretation of Conf. Intervals
Class Example 13
Q1: Summarize Coverage
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Interpretation of Conf. Intervals
Class Example 13
Q1: Summarize Coverage
94% > 90%
(since sd too small)
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Interpretation of Conf. Intervals
Class Example 13
Q2: Summarize Coverage
77% < 90%
(since too
variable)
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Interpretation of Conf. Intervals
Class Example 13
Q3: Summarize Coverage
77% < 90%
(since too
biased)
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Interpretation of Conf. Intervals
Class Example 13
Q4: Summarize Coverage
87% ≈ 90%
(seems OK?)
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Interpretation of Conf. Intervals
Class Example 13
Simulate from
Binomial
87% ≈ 90%
(shows within
expected
range)
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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”
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Interpretation of Conf. Intervals
HW: 6.20 ($1260, $1540), 6.21
6.28 (but use Excel & make histogram)
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Research Corner
Another SiZer analysis:
British Incomes Data
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Research Corner
Another SiZer analysis:
British Incomes Data
o Annual Survey (1985)
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Research Corner
Another SiZer analysis:
British Incomes Data
o Annual Survey (1985)
o Done in Great Britain
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Research Corner
Another SiZer analysis:
British Incomes Data
o Annual Survey (1985)
o Done in Great Britain
o Variable of Interest:
Family Income
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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?
![Page 70: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/70.jpg)
Research Corner
British Incomes Data
SiZer Results:
1 bump at coarse scale
(expected)
![Page 71: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/71.jpg)
Research Corner
British Incomes Data
SiZer Results:
1 bump at coarse scale
2 bumps at medium scale
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Research Corner
British Incomes Data
SiZer Results:
1 bump at coarse scale
2 bumps at medium scale
(Quite a radical
statement)
![Page 73: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/73.jpg)
Research Corner
British Incomes Data
SiZer Results:
1 bump at coarse scale
2 bumps at medium scale
Finer scale bumps not
statistically significant
![Page 74: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/74.jpg)
Research Corner
British Incomes Data
2 bumps at medium scale
Usual models for Incomes
(one bump only)
![Page 75: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/75.jpg)
Research Corner
British Incomes Data
2 bumps at medium scale
Usual models for Incomes
(one bump only)
2 bumps were verified
![Page 76: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/76.jpg)
Research Corner
British Incomes Data
2 bumps at medium scale
Usual models for Incomes
(one bump only)
2 bumps were verified
(in PhD dissertation)
![Page 77: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/77.jpg)
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?
![Page 78: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/78.jpg)
Next time
• Add multiple year plots as well
In:
IncomesAllKDE.mpg
![Page 79: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/79.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
![Page 80: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/80.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
nNX /,0~
![Page 81: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/81.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
Usually σ is unknown
nNX /,0~
![Page 82: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/82.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
Usually σ is unknown, so replace with an estimate, s.
nNX /,0~
![Page 83: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/83.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
Usually σ is unknown, so replace with an estimate, s.
For n large, should be “OK”
nNX /,0~
![Page 84: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/84.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
Usually σ is unknown, so replace with an estimate, s.
For n large, should be “OK”, but what about:
• n small?
nNX /,0~
![Page 85: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/85.jpg)
Deeper look at Inference
Recall: “inference” = CIs and Hypo Tests
Main Issue: In sampling distribution
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”?
nNX /,0~
![Page 86: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/86.jpg)
Unknown SD
Goal: Account for “extra variability in the
s ≈ σ approximation”
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Unknown SD
Goal: Account for “extra variability in the
s ≈ σ approximation”
Mathematics: Assume individual ,~ NX i
![Page 88: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/88.jpg)
Unknown SD
Goal: Account for “extra variability in the
s ≈ σ approximation”
Mathematics: Assume individual
I.e.
• Data have mound shaped histogram
,~ NX i
![Page 89: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/89.jpg)
Unknown SD
Goal: Account for “extra variability in the
s ≈ σ approximation”
Mathematics: Assume individual
I.e.
• Data have mound shaped histogram
• Recall averages generally normal
,~ NX i
![Page 90: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/90.jpg)
Unknown SD
Goal: Account for “extra variability in the
s ≈ σ approximation”
Mathematics: Assume individual
I.e.
• Data have mound shaped histogram
• Recall averages generally normal
• But now must focus on individuals
,~ NX i
![Page 91: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/91.jpg)
Unknown SD
Then nNX /,~
![Page 92: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/92.jpg)
Unknown SD
Then
So can write:
nNX /,~
1,0~ N
n
X
![Page 93: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/93.jpg)
Unknown SD
Then
So can write:
(recall: standardization (Z-score)
idea)
nNX /,~
1,0~ N
n
X
![Page 94: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/94.jpg)
Unknown SD
Then
So can write:
(recall: standardization (Z-score)
idea,
used in an important way here)
nNX /,~
1,0~ N
n
X
![Page 95: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/95.jpg)
Unknown SD
Then
So can write:
Replace
nNX /,~
1,0~ N
n
X
![Page 96: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/96.jpg)
Unknown SD
Then
So can write:
Replace by
nNX /,~
1,0~ N
n
X
s
![Page 97: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/97.jpg)
Unknown SD
Then
So can write:
Replace by , then
nNX /,~
1,0~ N
n
X
sn
sX
![Page 98: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/98.jpg)
Unknown SD
Then
So can write:
Replace by , then
has a distribution named
nNX /,~
1,0~ N
n
X
sn
sX
![Page 99: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/99.jpg)
Unknown SD
Then
So can write:
Replace by , then
has a distribution named:
“t-distribution with n-1 degrees of freedom”
nNX /,~
1,0~ N
n
X
sn
sX
![Page 100: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/100.jpg)
t - Distribution
Notes:
1. n is a parameter
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t - Distribution
Notes:
1. n is a parameter (like ),,, p
![Page 102: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/102.jpg)
t - Distribution
Notes:
1. n is a parameter (like )
(Recall: these index families
of probability distributions)
,,, p
![Page 103: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/103.jpg)
t - Distribution
Notes:
1. n is a parameter (like ) that controls “added variability that comes from the s ≈ σ approximation”
,,, p
![Page 104: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/104.jpg)
t - Distribution
Notes:
1. n is a parameter (like ) 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
,,, p
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t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
![Page 106: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/106.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
![Page 107: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/107.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
![Page 108: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/108.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread:
- Lower Peak
![Page 109: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/109.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread:
- Lower Peak
- Fatter Tails
![Page 110: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/110.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
smaller 5%-tile
![Page 111: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/111.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
smaller 5%-tile
larger 99%-tile
![Page 112: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/112.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
t is more spread
Makes sense,
since s ≈ σ
more variation
![Page 113: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/113.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 3
All effects are
magnified
Since s ≈ σ approx
gets worse
![Page 114: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/114.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 1
Extreme Case
Have terrible
s ≈ σ approx
![Page 115: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/115.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 7
Now try larger d.f.
![Page 116: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/116.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 14
All approximations
are better
![Page 117: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/117.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 25
Even better
![Page 118: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/118.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 25
Even better
- Densities almost
on top
![Page 119: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/119.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 25
Even better
- Densities almost
on top
- Quantiles very
close
![Page 120: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/120.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 100
Hard to see any
difference
![Page 121: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/121.jpg)
t - Distribution
Compare N(0,1) distribution, to t-distribution,
d.f. = 100
Hard to see any
difference
Since excellent
s ≈ σ approx
![Page 122: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/122.jpg)
t - Distribution
Notes:
2. Careful: set “degrees of freedom” =
= n – 1
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t - Distribution
Notes:
2. Careful: set “degrees of freedom” =
= n – 1 (not n)
![Page 124: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/124.jpg)
t - Distribution
Notes:
2. Careful: set “degrees of freedom” =
= n – 1 (not n)
• Easy to forget later
![Page 125: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/125.jpg)
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
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t - Distribution
Notes:
3. Must work with standardized version of
X
![Page 127: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/127.jpg)
t - Distribution
Notes:
3. Must work with standardized version of
i.e.
nsX X
![Page 128: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/128.jpg)
t - Distribution
Notes:
3. Must work with standardized version of
i.e.
(will affect how we compute probs….)
nsX X
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t - Distribution
Notes:
3. Must work with standardized version of
i.e.
• No longer can plug mean and SD
into EXCEL formulas
nsX X
![Page 130: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/130.jpg)
t - Distribution
Notes:
3. Must work with standardized version of
i.e.
• No longer can plug mean and SD
into EXCEL formulas
• In text standardization was already done
nsX X
![Page 131: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/131.jpg)
t - Distribution
Notes:
3. Must work with standardized version of
i.e.
• No longer can plug mean and SD
into EXCEL formulas
• In text standardization was already done,
since used in Normal table calc’ns
nsX X
![Page 132: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/132.jpg)
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
![Page 133: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/133.jpg)
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST
![Page 134: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/134.jpg)
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST & TINV
![Page 135: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/135.jpg)
t - Distribution
Notes:
4. Calculate t probs (e.g. areas & cutoffs),
using TDIST & TINV
Caution: these are set up differently from NORMDIST & NORMINV
![Page 136: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/136.jpg)
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 14http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg14.xls
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t - Distribution
Class Example 14:
![Page 138: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/138.jpg)
t - Distribution
Class Example 14:
Calculate Upper Prob
![Page 139: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/139.jpg)
t - Distribution
Class Example 14:
Calculate Upper Prob
![Page 140: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/140.jpg)
t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
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t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
(Check TDIST menu)
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t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
- cutoff
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t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
- cutoff
- d. f.
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t - Distribution
Class Example 14:
Calculate Upper Prob
Using TDIST
- cutoff
- d. f.
- upper prob. only
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t - Distribution
Class Example 14:
Careful: opposite from NORMDIST
Using TDIST
- cutoff
- d. f.
- upper prob. only
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t - Distribution
Class Example 14:
Careful: opposite from NORMDIST
use upper
Using TDIST
- cutoff
- d. f.
- upper prob. only
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t - Distribution
Class Example 14:
Careful: opposite from NORMDIST
use upper, NOT lower probs
Using TDIST
- cutoff
- d. f.
- upper prob. only
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t - Distribution
Class Example 14:
To compute lower prob
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t - Distribution
Class Example 14:
To compute lower prob
Use “1 – trick”, i.e. Not Rule of probability
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t - Distribution
Class Example 14:
How about upper prob of negative?
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t - Distribution
Class Example 14:
How about upper prob of negative?
Give it a try
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t - Distribution
Class Example 14:
How about upper prob of negative?
Give it a try
Get an error message in response
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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)
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t - Distribution
Class Example 14:
Reason:
TDIST tuned for 2-tailed
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t - Distribution
Class Example 14:
Reason:
TDIST tuned for 2-tailed
(where need cutoff > 0)
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t - Distribution
Class Example 14:
Reason:
TDIST tuned for 2-tailed
(where need cutoff > 0)
(correct version for CIs and H. tests)
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t - Distribution
Class Example 14:
Approach:
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t - Distribution
Class Example 14:
Approach:
Use “1 – trick”
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t - Distribution
Class Example 14:
Approach:
Use “1 – trick”
(to write as prob. can compute)
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t - Distribution
Class Example 14:
For Two-Tailed Prob
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t - Distribution
Class Example 14:
For Two-Tailed Prob
TDIST is very convenient
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t - Distribution
Class Example 14:
For Two-Tailed Prob
TDIST is very convenient
(much better than NORMDIST)
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t - Distribution
Class Example 14:
For Interior Prob
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t - Distribution
Class Example 14:
For Interior Prob
Use “1 – trick”
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t - Distribution
Class Example 14:
For Interior Prob
Use “1 – trick”
TDIST again very convenient
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t - Distribution
Class Example 14:
For Interior Prob
Use “1 – trick”
TDIST again very convenient
(again better than NORMDIST)
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t - Distribution
Class Example 14:
Now try increasing d.f.
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t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
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t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
But converges for larger n
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t - Distribution
Class Example 14:
Now try increasing d.f.
Big difference for small n
But converges for larger n
To Normal(0,1)
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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)
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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)
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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)
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And now for something completely different
“Thinking Outside the Box”
Also Called:“Lateral Thinking”
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And now for something completely different
Find the word or simple phrase suggested:
death ..... life
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And now for something completely different
Find the word or simple phrase suggested:
death ..... life
life after death
![Page 177: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/177.jpg)
And now for something completely different
Find the word or simple phrase suggested:
ecnalg
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And now for something completely different
Find the word or simple phrase suggested:
ecnalg
backward
glance
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And now for something completely different
Find the word or simple phrase suggested:
He's X himself
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And now for something completely different
Find the word or simple phrase suggested:
He's X himself
He's by himself
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And now for something completely different
Find the word or simple phrase suggested:
THINK
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And now for something completely different
Find the word or simple phrase suggested:
THINK
think big ! !
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And now for something completely different
Find the word or simple phrase suggested:
ababaaabbbbaaaabbbb ababaabbaaabbbb. .
![Page 184: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/184.jpg)
And now for something completely different
Find the word or simple phrase suggested:
ababaaabbbbaaaabbbb ababaabbaaabbbb. .
long time no 'C'
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t - Distribution
Class Example 14:
Next explore TINV
(Inverse function)
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t - Distribution
Class Example 14:
Next explore TINV
(Inverse function)
(Given cutoff, find area)
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t - Distribution
Class Example 14:
Next explore TINV
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t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area)
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t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area) & d.f.
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t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area) & d.f., find cutoff
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t - Distribution
Class Example 14:
Next explore TINV
Given prob. (area) & d.f., find cutoff
(next think carefully about interpretation)
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Now invert this,
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Now invert this, i.e. given prob.
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Now invert this, i.e. given prob., find cutoff
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
For same d.f.
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
For same d.f., use resulting prob. as input
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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
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding?
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding? Try exact value
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t - Distribution
Class Example 14:
Next explore TINV
Recall TDIST e.g. from above:
Maybe due to rounding? Try exact value
![Page 202: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/202.jpg)
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
![Page 203: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/203.jpg)
t - Distribution
Class Example 14:
Next explore TINV
Reason for inconsistency:
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t - Distribution
Class Example 14:
Next explore TINV
Reason for inconsistency:
Works via 2-tailed
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t - Distribution
Class Example 14:
Next explore TINV
Reason for inconsistency:
Works via 2-tailed, not 1-tailed, probability
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t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
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t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
![Page 208: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/208.jpg)
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
![Page 209: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/209.jpg)
t - Distribution
Class Example 14: Explore TINV
Works via 2-tailed, not 1-tailed, probability
Check by inverting 2-tailed answer above:
Get:
![Page 210: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/210.jpg)
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
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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
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EXCEL Functions
Summary:
Normal:
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EXCEL Functions
Summary:
Normal:
plug in: get out:
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EXCEL Functions
Summary:
Normal:
plug in: get out:
NORMDIST: cutoff
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EXCEL Functions
Summary:
Normal:
plug in: get out:
NORMDIST: cutoff area
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EXCEL Functions
Summary:
Normal:
plug in: get out:
NORMDIST: cutoff area
NORMINV: area
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EXCEL Functions
Summary:
Normal:
plug in: get out:
NORMDIST: cutoff area
NORMINV: area cutoff
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EXCEL Functions
Summary:
Normal:
plug in: get out:
NORMDIST: cutoff area
NORMINV: area cutoff
(but TDIST is set up really differently)
![Page 219: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/219.jpg)
EXCEL Functionst distribution:
1 tail:
![Page 220: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/220.jpg)
EXCEL Functionst distribution:
1 tail:
plug in: get out:
![Page 221: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/221.jpg)
EXCEL Functionst distribution:
1 tail:
plug in: get out:
TDIST: cutoff
![Page 222: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/222.jpg)
EXCEL Functionst distribution:
1 tail:
plug in: get out:
TDIST: cutoff area
![Page 223: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/223.jpg)
EXCEL Functionst distribution:
1 tail:
plug in: get out:
TDIST: cutoff area
EXCEL notes: - no explicit inverse
![Page 224: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/224.jpg)
EXCEL Functionst distribution:
1 tail:
plug in: get out:
TDIST: cutoff area
EXCEL notes: - no explicit inverse
- backwards from Normal…
![Page 225: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/225.jpg)
EXCEL Functions
t distribution:
2 tail:
![Page 226: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/226.jpg)
EXCEL Functions
t distribution:
2 tail:
plug in: get out:
![Page 227: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/227.jpg)
EXCEL Functions
t distribution:
2 tail:
plug in: get out:
TDIST: cutoff
![Page 228: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/228.jpg)
EXCEL Functions
t distribution:
Area
2 tail:
plug in: get out:
TDIST: cutoff area
![Page 229: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/229.jpg)
EXCEL Functions
t distribution:
Area
2 tail:
plug in: get out:
TDIST: cutoff area
TINV: area
![Page 230: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/230.jpg)
EXCEL Functions
t distribution:
Area
2 tail:
plug in: get out:
TDIST: cutoff area
TINV: area cutoff
![Page 231: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/231.jpg)
EXCEL Functions
t distribution:
Area
2 tail:
plug in: get out:
TDIST: cutoff area
TINV: area cutoff
(EXCEL note: this one has the inverse)
![Page 232: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/232.jpg)
EXCEL Functions
Note: when need to invert the 1-tail TDIST,
Use twice the area.
![Page 233: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/233.jpg)
EXCEL Functions
Note: when need to invert the 1-tail TDIST,
Use twice the area.
Area = A
![Page 234: Administrative Matters Midterm II Results Take max of two midterm scores:](https://reader035.fdocuments.in/reader035/viewer/2022062309/5697c0241a28abf838cd4b8e/html5/thumbnails/234.jpg)
EXCEL Functions
Note: when need to invert the 1-tail TDIST,
Use twice the area.
Area = A Area = 2 A
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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)