last time binomial distribution –excel computation political polls –strength of evidence...
TRANSCRIPT
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Last Time
• Binomial Distribution– Excel Computation
• Political Polls– Strength of evidence
• Hypothesis Testing– Yes – No Questions
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Administrative Matter
• Midterm I, coming Tuesday, Feb. 24
(will say more later)
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Reading In Textbook
Approximate Reading for Today’s Material:
Pages 488-491, 317-318
Approximate Reading for Next Class:
Pages 261-262, 9-14, 270-276, 30-34
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Haircut?
Why?
Website:
http://www.time.com/time/health/article/0,8599,1733719,00.html
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Haircut?
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Hypothesis Testing
Example: Suppose surgery cures (a certain
type of) cancer 60% of time
Q: is eating apricot pits a more effective cure?
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Hypothesis Testing
E.g. Pits vs. Surgery
Let p be “cure rate” of pits
(i.e. proportion of people cured)
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Hypothesis Testing
E.g. Pits vs. Surgery
Let p be “cure rate” of pits
(H0 & H1? New method needs to
“prove it’s worth”
so put burden of proof on it)
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Hypothesis Testing
E.g. Pits vs. Surgery
Let p be “cure rate” of pits
H0: p < 0.6 vs. H1: p ≥ 0.6
Recall cure rate of surgery
(competing treatment)
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Hypothesis Testing
E.g. Pits vs. Surgery
Let p be “cure rate” of pits
H0: p < 0.6 vs. H1: p ≥ 0.6
(OK to be sure of “at least as good”,
since pits nicer than surgery)
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were
cured by pits
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were
cured by pits
I.e.: “best guess about p” is:
733.0ˆ1511 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were
cured by pits
I.e.: “best guess about p” is:
6.0733.0ˆ1511 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were
cured by pits
I.e.: “best guess about p” is:
6.0733.0ˆ1511 n
Xp
Looks Better?
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were
cured by pits
I.e.: “best guess about p” is:
But is it conclusive?
6.0733.0ˆ1511 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were
cured by pits
I.e.: “best guess about p” is:
But is it conclusive?
6.0733.0ˆ1511 n
Xp
Or just due to sampling variation?
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Hypothesis Testing
Approach: Define
“p-value” =
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Hypothesis Testing
Approach: Define
“p-value” = “observed significance level”
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Hypothesis Testing
Approach: Define
“p-value” = “observed significance level”
= “significance probability”
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Hypothesis Testing
Approach: Define
“p-value” = “observed significance level”
= “significance probability”
= P[seeing something as
unusual as 11 | H0 is
true]
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is
true]
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is
true]
Note: for
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is
true]
Note: for could use “X/n = 0.733”
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is
true]
Note: for could use “X/n = 0.733”,
but this depends too much on n
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is true]
Note: for could use “X/n = 0.733”,
but this depends too much on n
(look at example illustrating this)
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Class Example 4
For X ~ Bi(n,0.6):n P(X/n = 0.6) P(X/n >= 0.6)
5 0.346 0.31710 0.251 0.36730 0.147 0.422
100 0.081 0.457300 0.047 0.475
1000 0.026 0.4863000 0.015 0.492
10000 0.008 0.496
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Class Example 4
For X ~ Bi(n,0.6):
Computed using
Excel:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg4.xls
n P(X/n = 0.6) P(X/n >= 0.6)
5 0.346 0.31710 0.251 0.36730 0.147 0.422
100 0.081 0.457300 0.047 0.475
1000 0.026 0.4863000 0.015 0.492
10000 0.008 0.496
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Class Example 4
For X ~ Bi(n,0.6):
Note: these go to 0,
even at “most likely
value”
n P(X/n = 0.6) P(X/n >= 0.6)
5 0.346 0.31710 0.251 0.36730 0.147 0.422
100 0.081 0.457300 0.047 0.475
1000 0.026 0.4863000 0.015 0.492
10000 0.008 0.496
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Class Example 4
For X ~ Bi(n,0.6):
Note: these go to 0,
even at “most likely
value”
So “small” is
not conclusive
n P(X/n = 0.6) P(X/n >= 0.6)
5 0.346 0.31710 0.251 0.36730 0.147 0.422
100 0.081 0.457300 0.047 0.475
1000 0.026 0.4863000 0.015 0.492
10000 0.008 0.496
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Class Example 4
For X ~ Bi(n,0.6):
But for these
“small”
is conclusive
n P(X/n = 0.6) P(X/n >= 0.6)
5 0.346 0.31710 0.251 0.36730 0.147 0.422
100 0.081 0.457300 0.047 0.475
1000 0.026 0.4863000 0.015 0.492
10000 0.008 0.496
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Class Example 4
For X ~ Bi(n,0.6):
But for these
“small”
is conclusive
(so use range,
not value)
n P(X/n = 0.6) P(X/n >= 0.6)
5 0.346 0.31710 0.251 0.36730 0.147 0.422
100 0.081 0.457300 0.047 0.475
1000 0.026 0.4863000 0.015 0.492
10000 0.008 0.496
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing 11 or more
unusual | H0 is true]
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Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing 11 or more
unusual | H0 is true]
So use:
= P[X ≥ 11 | H0 is true]
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true]
![Page 36: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/36.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true]
What to use here?
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true]
What to use here?
Recall: H0: p < 0.6
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true]
What to use here?
Recall: H0: p < 0.6
How does P[X ≥ 11 | p] depend on p?
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Hypothesis Testing
How does P[X ≥ 11 | p] depend on p?
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Hypothesis Testing
How does P[X ≥ 11 | p] depend on p?
Calculated in Class EG 4b:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg4.xls
p P(X >= 11|p)
0.2 0.000
0.3 0.001
0.4 0.009
0.5 0.059
0.6 0.217
0.7 0.515
0.8 0.836
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Hypothesis Testing
How does P[X ≥ 11 | p] depend on p?
Bigger assumed p
goes with
Bigger Probability
i.e. less conclusive
p P(X >= 11|p)
0.2 0.000
0.3 0.001
0.4 0.009
0.5 0.059
0.6 0.217
0.7 0.515
0.8 0.836
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
So, to be “sure” of conclusion, use largest
available value of P[X ≥ 11 | p]
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
So, to be “sure” of conclusion, use largest
available value of P[X ≥ 11 | p]
Thus, define:
“p-value” = P[X ≥ 11 | p = 0.6]
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Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
So, to be “sure” of conclusion, use largest
available value of P[X ≥ 11 | p]
Thus, define:
“p-value” = P[X ≥ 11 | p = 0.6]
(since “=” gives safest result)
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]
Generally: use
= P[seeing something as
unusual as X = 11 | H0 is
true]
![Page 48: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/48.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]
Generally: use
= P[seeing something as
unusual as X = 11 | H0 is
true]
Here use boundary between H0 & H1
![Page 49: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/49.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]
Generally: use
= P[seeing something as
unusual as X = 11 | H0 is true]
Here use boundary between H0 & H1
(above e.g. p = 0.6)
![Page 50: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/50.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]
Now calculate numerical value
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]
Now calculate numerical value
(already done above,
Class EG 4)
![Page 52: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/52.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Now calculate numerical value
(already done above,
Class EG 4)
![Page 53: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/53.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Now calculate numerical value
(already done above,
Class EG 4)
How to interpret?
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error
when H0 is rejected
![Page 55: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/55.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error
when H0 is rejected
(i.e. when conclusion is made)
![Page 56: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/56.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error
when H0 is rejected
(i.e. when conclusion is made)
(based on available evidence)
![Page 57: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/57.jpg)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error
when H0 is rejected
(i.e. when conclusion is made)
(based on available evidence)
When p-value is small, it is safe to make a
firm conclusion
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Hypothesis Testing
For small p-value, safe to make firm conclusion
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Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
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Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
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Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
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Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
Reject H0 when p-value < 0.05
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Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
Reject H0 when p-value < 0.05
(just an agreed upon value,
but very widely used)
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Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
Reject H0 when p-value < 0.05
(but sometimes want different values,
e.g. your airplane is safe to fly)
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Hypothesis Testing
Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
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Hypothesis Testing
Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
Terminology: say results are “statistically
significant”, when this happens
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Hypothesis Testing
Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
Terminology: say results are “statistically
significant”, when this happens
Sometimes specify a value α
Greek letter “alpha”
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Hypothesis Testing
Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
Terminology: say results are “statistically
significant”, when this happens
Sometimes specify a value α
as the cutoff (different from 0.05)
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Hypothesis Testing
Approach 2: “Gray Level”
Idea: allow “shades of conclusion”
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Hypothesis Testing
Approach 2: “Gray Level”
Idea: allow “shades of conclusion”
e.g. Do p-val = 0.049 and p-val = 0.051
represent very different levels of evidence?
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Hypothesis Testing
Approach 2: “Gray Level”
Idea: allow “shades of conclusion”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong
evidence
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Hypothesis Testing
Approach 2: “Gray Level”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong
evidence
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Hypothesis Testing
Approach 2: “Gray Level”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong
evidence
stronger when closer to 0.01
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Hypothesis Testing
Approach 2: “Gray Level”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong evidence
stronger when closer to 0.01
weaker when closer to 0.1
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
Bottom Line:
Yes-No: can not reject H0, since
0.217 > 0.05
i.e. no firm evidence pits better than
surgery
Gray level: not much indicated
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than
surgery
Gray level: not much indicated
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than
surgery
Gray level: not much indicated
Practical Issue: since 73% = observed rate for
pits > 60% (surgery),
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than
surgery
Gray level: not much indicated
Practical Issue: since 73% = observed rate for
pits > 60% (surgery), may want to gather
more data
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Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than
surgery
Gray level: not much indicated
Practical Issue: since 73% = observed rate for
pits > 60% (surgery), may want to gather
more data, might show value of pits
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Research Corner
Medical Imaging – Another Fun ExampleMedical Imaging – Another Fun Example
Cornea DataCornea Data
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Research Corner
Medical Imaging – Another Fun ExampleMedical Imaging – Another Fun Example
Cornea DataCornea Data
Cornea = Outer surface Cornea = Outer surface
of eyeof eye
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Research Corner
Medical Imaging – Another Fun ExampleMedical Imaging – Another Fun Example
Cornea DataCornea Data
Cornea = Outer surface Cornea = Outer surface
of eyeof eye
““Curvature” important toCurvature” important to
visionvision
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Research Corner
Medical Imaging – Another Fun ExampleMedical Imaging – Another Fun Example
Cornea DataCornea Data
Cornea = Outer surface Cornea = Outer surface
of eyeof eye
““Curvature” important toCurvature” important to
visionvision
Study Study heat map heat map showingshowing
curvaturecurvature
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Research Corner
Cornea DataCornea Data
Heat map Heat map shows curvatureshows curvature
Each image is one personEach image is one person
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Research Corner
Cornea DataCornea Data
Heat map Heat map shows curvatureshows curvature
Each image is one personEach image is one person
Understand “populationUnderstand “population
variation”?variation”?
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Research Corner
Cornea DataCornea Data
Heat map Heat map shows curvatureshows curvature
Each image is one personEach image is one person
Understand “populationUnderstand “population
variation”?variation”?
(too messy for brain(too messy for brain
to summarize)to summarize)
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Research Corner
Cornea DataCornea Data
Approach: PrincipalApproach: Principal
Component AnalysisComponent Analysis
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Research Corner
Cornea DataCornea Data
Approach: PrincipalApproach: Principal
Component AnalysisComponent Analysis
Idea: follow “direction” inIdea: follow “direction” in
image space, image space,
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Research Corner
Cornea DataCornea Data
Approach: PrincipalApproach: Principal
Component AnalysisComponent Analysis
Idea: follow “direction” inIdea: follow “direction” in
image space, that highlightsimage space, that highlights
population featurespopulation features
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Research Corner
Cornea DataCornea Data
Population featuresPopulation features
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Research Corner
Cornea DataCornea Data
Population featuresPopulation features
• Overall curvatureOverall curvature
(hot – cold)(hot – cold)
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Research Corner
Cornea DataCornea Data
Population featuresPopulation features
• Overall curvatureOverall curvature
(hot – cold)(hot – cold)
• With the rule astigmatismWith the rule astigmatism
(figure 8 pattern)(figure 8 pattern)
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Research Corner
Cornea DataCornea Data
Population featuresPopulation features
• Overall curvatureOverall curvature
(hot – cold)(hot – cold)
• With the rule astigmatismWith the rule astigmatism
(figure 8 pattern)(figure 8 pattern)
• CorrelationCorrelation
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
(cured by pits)
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
(cured by pits)
(recall above saw 11 / 25 not conclusive,
so now suppose stronger evidence)
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So 1513ˆ n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So %7.86ˆ1513 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So
(more conclusive than before)
%60%7.86ˆ1513 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So
(more conclusive than before)
(how much stronger is the evidence?)
%60%7.86ˆ1513 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So
p-value = P[ X ≥ 13 | p = 0.6]
%7.86ˆ1513 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
%7.86ˆ1513 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
So
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
Calculated similar to above:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg4.xls
%7.86ˆ1513 n
Xp
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
Conclusions:
Yes-No: 0.027 < 0.05, so can reject H0 and
make firm conclusion pits are better
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Hypothesis Testing
H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
Conclusions:
Yes-No: 0.027 < 0.05, so can reject H0 and
make firm conclusion pits are better
Gray Level: Strong case, nearly very strong that
pits are better
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Hypothesis Testing
In General:
p-value = P[what was seen,
or more conclusive | at
boundary between
H0 & H1]
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Hypothesis Testing
In General:
p-value = P[what was seen,
or more conclusive | at
boundary between
H0 & H1]
(will use this throughout the course,
well beyond Binomial distributions)
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Hypothesis Testing
HW C14: Answer from both gray-level and
yes-no viewpoints:
(a) A TV ad claims that less than 40% of
people prefer Brand X. Suppose 7 out of
10 randomly selected people prefer Brand
X. Should we dispute the claim? (p-value
= 0.055)
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Hypothesis Testing
HW C14: Answer from both gray-level and
yes-no viewpoints:
(b) 80% of the sheet metal we buy from
supplier A meets our specs. Supplier B
sends us 12 shipments, and 11 meet our
specs. Is it safe to say the quality of B is
higher? (p-value = 0.275)
![Page 110: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/110.jpg)
Warning
Avoid the “Excel Twiddle Trap”
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Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
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Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Find what Excel needs:
![Page 113: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/113.jpg)
Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Find what Excel needs:
Number_s: 7
Trials: 10
Probability_s: 0.4
Cumulative: true
(plug in)
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Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Check given answer
(0.055)
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Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Check given answer
(0.055)
Way off!
![Page 116: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/116.jpg)
Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
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Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
Still off, how about
the “> vs. ≥” issue?
![Page 118: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/118.jpg)
Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
Still off, how about
the “> vs. ≥” issue?
try replacing 7 by 6?
![Page 119: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/119.jpg)
Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
Still off, how about
the “> vs. ≥” issue?
try replacing 7 by 6? Yes!
![Page 120: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/120.jpg)
Warning
Avoid the “Excel Twiddle Trap”:
• Can solve HW OK
![Page 121: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/121.jpg)
Warning
Avoid the “Excel Twiddle Trap”:
• Can solve HW OK
• But not on exam
– No numerical answer given
– No interaction with Excel
![Page 122: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/122.jpg)
Warning
Avoid the “Excel Twiddle Trap”:
• Can solve HW OK
• But not on exam
– No numerical answer given
– No interaction with Excel
• Real Goal: Understanding Principles
![Page 123: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/123.jpg)
And now for something completely different
Lateral Thinking: What is the phrase?
![Page 124: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/124.jpg)
And now for something completely different
Lateral Thinking: What is the phrase?
Card
Shark
![Page 125: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/125.jpg)
And now for something completely different
Lateral Thinking:What is the phrase?
![Page 126: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/126.jpg)
And now for something completely different
Lateral Thinking:What is the phrase?
Knight Mare
![Page 127: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/127.jpg)
And now for something completely different
Lateral Thinking:What is the phrase?
![Page 128: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/128.jpg)
And now for something completely different
Lateral Thinking:What is the phrase?
Gator Aide
![Page 129: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/129.jpg)
Hypothesis Testing
In General:
p-value = P[what was seen,
or more conclusive | at
boundary between
H0 & H1]
![Page 130: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/130.jpg)
Hypothesis Testing
In General:
p-value = P[what was seen,
or more conclusive | at
boundary between
H0 & H1]
Caution: more conclusive requires careful
interpretation
![Page 131: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/131.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
![Page 132: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/132.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
Reason: Need to decide between
1 - sided Hypotheses
![Page 133: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/133.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
Reason: Need to decide between
1 - sided Hypotheses, like
H0 : p < vs. H1: p ≥
some given numerical value
![Page 134: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/134.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
Reason: Need to decide between
1 - sided Hypotheses, like
H0 : p < vs. H1: p ≥
And 2 - sided Hypotheses
![Page 135: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/135.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
Reason: Need to decide between
1 - sided Hypotheses, like
H0 : p < vs. H1: p ≥
And 2 - sided Hypotheses, like
H0 : p = vs. H1: p ≠
![Page 136: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/136.jpg)
Hypothesis Testing
2 - sided Hypotheses, like
H0 : p = vs. H1: p ≠
Note: Can never have H1: p =
![Page 137: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/137.jpg)
Hypothesis Testing
2 - sided Hypotheses, like
H0 : p = vs. H1: p ≠
Note: Can never have H1: p = ,
since can’t tell for sure between
and + 0.000001
![Page 138: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/138.jpg)
Hypothesis Testing
2 - sided Hypotheses, like
H0 : p = vs. H1: p ≠
Note: Can never have H1: p = ,
since can’t tell for sure between
and + 0.000001
(Recall: H1 has burden of proof)
![Page 139: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/139.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
1 - sided Hypotheses & 2 - sided
Hypotheses
![Page 140: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/140.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
1 - sided Hypotheses & 2 - sided
Hypotheses
(important choice will need to make a lot)
![Page 141: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/141.jpg)
Hypothesis Testing
Caution: more conclusive requires careful
interpretation
1 - sided Hypotheses & 2 - sided
Hypotheses
Useful Rule: set up 2-sided when problem
uses words like “equal” or “different”
![Page 142: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/142.jpg)
Hypothesis Testing
e.g. a slot machine
• Gambling device
![Page 143: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/143.jpg)
Hypothesis Testing
e.g. a slot machine
• Gambling device
• Players put money in
![Page 144: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/144.jpg)
Hypothesis Testing
e.g. a slot machine
• Gambling device
• Players put money in
• With (small) probability, win a “jackpot”
(of quite a lot more money)
![Page 145: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/145.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
![Page 146: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/146.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
(in real life, focus is on “return rate”)
![Page 147: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/147.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
(in real life, focus is on “return rate”)
(since people enjoy fewer, but bigger jackpots)
![Page 148: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/148.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
(in real life, focus is on “return rate”)
(since people enjoy fewer, but bigger jackpots)
(but usually no signs,
since return rate is < 0)
![Page 149: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/149.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any.
![Page 150: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/150.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any.
Can I conclude sign is false?
![Page 151: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/151.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any.
Can I conclude sign is false?
(& thus have grounds for complaint,
or is this a reasonable occurrence?)
![Page 152: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/152.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win]
![Page 153: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/153.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win]
(usual approach: give unknowns a
name, so can work with)
![Page 154: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/154.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
![Page 155: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/155.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
![Page 156: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/156.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
(set up as H0, the point want to disprove)
![Page 157: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/157.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
![Page 158: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/158.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says “Win
30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
(“false” means don’t win 30% of time,
so go 2-sided)
![Page 159: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/159.jpg)
Hypothesis Testing
Aside (similar to above):
• Can never set up H0: p ≠ 0.3
![Page 160: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/160.jpg)
Hypothesis Testing
Aside (similar to above):
• Can never set up H0: p ≠ 0.3
• And then prove that p = 0.3
![Page 161: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/161.jpg)
Hypothesis Testing
Aside (similar to above):
• Can never set up H0: p ≠ 0.3
• And then prove that p = 0.3
• Since can’t handle gray area of hypo test
![Page 162: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/162.jpg)
Hypothesis Testing
Aside (similar to above):
• Can never set up H0: p ≠ 0.3
• And then prove that p = 0.3
• Since can’t handle gray area of hypo test
• E.g. can’t distinguish from p = 0.30001
![Page 163: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/163.jpg)
Hypothesis Testing
Aside (similar to above):
• Can never set up H0: p ≠ 0.3
• And then prove that p = 0.3
• Since can’t handle gray area of hypo test
• E.g. can’t distinguish from p = 0.30001
• Could always be “off a little bit”
![Page 164: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/164.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
![Page 165: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/165.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
(now test & see how weird X = 0 is, for p = 0.3)
![Page 166: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/166.jpg)
Hypothesis Testing
e.g. a slot machine bears a sign which says
“Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
![Page 167: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/167.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
![Page 168: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/168.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
(understand this by visualizing # line)
![Page 169: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/169.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
0 1 2 3 4 5 6
![Page 170: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/170.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
0 1 2 3 4 5 6
30% of 10, most likely when p = 0.3
i.e. least conclusive
![Page 171: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/171.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
0 1 2 3 4 5 6
so more conclusive includes
![Page 172: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/172.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
0 1 2 3 4 5 6
so more conclusive includes
but since 2-sided, also include
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Hypothesis Testing
Generally how to calculate?
0 1 2 3 4 5 6
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Hypothesis Testing
Generally how to calculate?
Observed Value
0 1 2 3 4 5 6
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Hypothesis Testing
Generally how to calculate?
Observed Value
Most Likely Value
0 1 2 3 4 5 6
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Hypothesis Testing
Generally how to calculate?
Observed Value
Most Likely Value
0 1 2 3 4 5 6
# spaces = 3
![Page 177: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/177.jpg)
Hypothesis Testing
Generally how to calculate?
Observed Value
Most Likely Value
0 1 2 3 4 5 6
# spaces = 3
so go 3 spaces in other
direct’n
![Page 178: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/178.jpg)
Hypothesis Testing
Result: More conclusive means
X ≤ 0 or X ≥ 6
0 1 2 3 4 5 6
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Hypothesis Testing
Result: More conclusive means
X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
![Page 180: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/180.jpg)
Hypothesis Testing
Result: More conclusive means
X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
![Page 181: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/181.jpg)
Hypothesis Testing
Result: More conclusive means
X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
= P[X ≤ 0] + (1 – P[X ≤ 5])
![Page 182: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/182.jpg)
Hypothesis Testing
Result: More conclusive means
X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
= P[X ≤ 0] + (1 – P[X ≤ 5])
= 0.076
![Page 183: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/183.jpg)
Hypothesis Testing
Result: More conclusive means
X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
= P[X ≤ 0] + (1 – P[X ≤ 5])
= 0.076
Excel result from:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg4.xls
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Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = 0.076
![Page 185: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/185.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = 0.076
Yes-No Conclusion: 0.076 > 0.05,
so not safe to conclude “P[win] = 0.3”
sign
is wrong, at level 0.05
![Page 186: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/186.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = 0.076
Yes-No Conclusion: 0.076 > 0.05,
so not safe to conclude “P[win] = 0.3”
sign
is wrong, at level 0.05
(10 straight losses is reasonably likely)
![Page 187: Last Time Binomial Distribution –Excel Computation Political Polls –Strength of evidence Hypothesis Testing –Yes – No Questions](https://reader037.vdocuments.us/reader037/viewer/2022103022/56649f555503460f94c79335/html5/thumbnails/187.jpg)
Hypothesis Testing
Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = 0.076
Yes-No Conclusion: 0.076 > 0.05,
so not safe to conclude “P[win] = 0.3”
sign
is wrong, at level 0.05
Gray Level Conclusion: in “fuzzy zone”,
some evidence, but not too strong