heads up! sept 22 – oct 4 probability perceived by many as a difficult topic get ready ahead of...
TRANSCRIPT
![Page 1: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/1.jpg)
Heads Up!Heads Up!Sept 22 – Oct 4Sept 22 – Oct 4
Probability
Perceived by many as a difficult topic
Get ready ahead of time
![Page 2: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/2.jpg)
Last Time:Last Time:
Least Squares Regression
(Simple Linear Regression)
Correlation
![Page 3: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/3.jpg)
In Least-Squares Regression:
XbYa
XX
YYXXb N
ii
N
iii
,
1
2
1
N
i
N
iii
N
i
N
ii
N
iiii
XXN
YXYXN
b
1
2
1
2
1 11
ComputationalFormula
![Page 4: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/4.jpg)
N
i
N
iii
N
i
N
ii
N
iiii
XXN
YXYXN
b
1
2
1
2
1 11Can wedo this?
X X Squared Y Y Squared XY86 7396 82.6 6822.76 7103.6
109.3 11946.49 112.6 12678.76 12307.1873.3 5372.89 70 4900 513180.6 6496.36 76.6 5867.56 6173.9686.6 7499.56 84 7056 7274.485.3 7276.09 86 7396 7335.883.3 6938.89 82.6 6822.76 6880.5878.6 6177.96 81.3 6609.69 6390.1892 8464 86.6 7499.56 7967.276 5776 75.3 5670.09 5722.8851 73344.24 837.6 71323.18 72286.7Totals:
![Page 5: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/5.jpg)
Calculating the Least Squares Regression Line contd.
2851)24.344,73(10
)6.837)(851()70.286,72(10
b
201,7244.442,733
6.797,712867,722
09.14.241,9
4.069,10
XbYa 910
85109.1
10
837
![Page 6: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/6.jpg)
10
10.9Slope is 1.09
Intercept is -9
You can’t see it in thisgraph
TRIAL = 1.09 PRACTICE - 9
RegressionEquation
![Page 7: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/7.jpg)
A view from further away….
0
20
40
60
80
100
120
0 50 100 150
X
Y
![Page 8: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/8.jpg)
Look at the residuals:
Residual Plot
-10
-5
0
5
10
0 20 40 60 80 100 120
X
Res
idu
als
We wanta shot-gun blast
shape, i.e.,a random blob
![Page 9: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/9.jpg)
Look at Residuals & Line Fit
ResidualPlot
Line FitPlot
Problem:Relationship is not linear
![Page 10: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/10.jpg)
Look at Residuals & Line Fit
ResidualPlot
Problem:Predictions are very precise for small predicted values,
but very unprecise for large predicted values. (Not good)
![Page 11: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/11.jpg)
1 2 3 4 5 6 7 8 9 10 11 12
Problem: Lurking (third) variables (?)
Here: Seasonal Trend?
Look at ResidualsResidual
Plot
![Page 12: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/12.jpg)
CorrelationHow strong is the linear relationship
between two variables X and Y?
Slope in regression ofstandardized variables
XX S
XXZ
YY S
YYZ
This slope tells meHow much a given change (in standardized units) of X
translates into a change (in standardized units) of Y
![Page 13: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/13.jpg)
CorrelationHow strong is the linear relationship
between two variables X and Y?
Correlation Coefficient
Y
iN
i X
iN S
YY
S
XXr
11
1
Computational Formula:
YX
N
ii
N
i
N
iiNii
SSN
YXYX
r)1(
11 1
1
![Page 14: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/14.jpg)
Properties of Correlation
• Symmetric Measure (You can exchange X and Y and get the same value)
• -1 ≤ r ≤ 1
• -1 is “perfect” negative correlation
• 1 is “perfect” positive correlation
• Not dependent on linear transformations of X and Y
• Measures linear relationship only
![Page 15: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/15.jpg)
X Z_X Y Z_Y Z_X Z_Y86 0.088816751 82.6 -0.10192166 -0.009052
109.3 2.388183737 112.6 2.533983252 6.051617673.3 -1.164486285 70 -1.20900172 1.407865980.6 -0.444083753 76.6 -0.62910264 0.279374386.6 0.148027918 84 0.021087239 0.003121585.3 0.019737056 86 0.196814233 0.003884583.3 -0.177633501 82.6 -0.10192166 0.018104778.6 -0.641454309 81.3 -0.2161442 0.138646692 0.680928421 86.6 0.249532331 0.169913776 -0.898036033 75.3 -0.74332518 0.6675328851 837.6 8.731009285.1 83.76 0.9701121
PRACTICE TRIALCASE 1 86 82.6CASE 2 109.3 112.6CASE 3 73.3 70CASE 4 80.6 76.6CASE 5 86.6 84CASE 6 85.3 86CASE 7 83.3 82.6CASE 8 78.6 81.3CASE 9 92 86.6CASE 10 76 75.3
Let’s try it out on our X = PRACTICE, Y = TRIAL
Data Set
Check this calculation at home!
![Page 16: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/16.jpg)
TodayToday
Finish Theory on RegressionFinish Theory on Regression
Pathologies and TrapsPathologies and Traps
in Linear Regression and Correlationin Linear Regression and Correlation
Relationships between Relationships between
Categorical VariablesCategorical Variables
![Page 17: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/17.jpg)
Regression on Standardized Variables
ii XY rZZ ˆ
0intercept, :slope1
11
N
iYXN iiZZr
X
i
Y
i
S
XXr
S
YY
ˆ
XXS
SrYY i
X
Yi ˆ
![Page 18: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/18.jpg)
ii XY rZZ ˆ
XXS
SrYY i
X
Yi ˆ
iX
Y
X
Yi X
S
SrX
S
SrYY ˆ
ii bXaY ˆ?
XbYaS
Srb
X
Y ,
![Page 19: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/19.jpg)
iiN bXaYYYY ˆ from ˆ,...,ˆ,ˆGiven 21
What is the variance of ? ˆ,...,ˆ,ˆ21 NYYY
22 bemust it that know We XSb
X
Y
S
Srb :know also We
2222 :Therefore YX SrSb
22
22
:Thus rS
Sb
Y
X
![Page 20: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/20.jpg)
22
22
:Thus rS
Sb
Y
X
Variance ofpredicted Y’s
Variance ofobserved Y’s
Proportion of Varianceof observed Y’s
that is accounted forby the regression
Proportion of Variance explained
![Page 21: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/21.jpg)
22
22
:Thus rS
Sb
Y
X
Proportion of Varianceof observed Y’s
that is accounted forby the regression
Proportion of Variance explained
Note: If you exchange X and Y in the regression, you find the same r and r squared
![Page 22: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/22.jpg)
Correlation only checks magnitude of
Linear Relationships!
It can happen that r=0, even though X and Y are highly related to each other!
Need to look at Scatter Plot and Residual Plot to make sure that you don’t miss an obvious relationship overlooked by linear regression!
![Page 23: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/23.jpg)
2XY Y = X-squared Line Fit Plot
-200
0
200
400
0 5 10 15 20
X
Y
How does a Linear RegressionModel approximate (for X=1,2,…,15)
Y = X-squared Residual Plot
-50
0
50
0 5 10 15 20
X
Res
idu
als
For these particular datathe regression
model finds
a = -45b = 16
The residuals have a systematic trend!!
This Linear Regressionis inappropriate!!
ii bXaY ˆ
![Page 24: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/24.jpg)
2XY How does a Linear RegressionModel approximate (for X=-8,-7,…,7,8)
For these particular datathe regression
model finds
a = 24b = 0
The residuals have a systematic trend!!
This Linear Regressionis inappropriate!!
Y = X_squared Line Fit Plot
0
50
100
-10 -5 0 5 10
X
Y
Y = X_squared Residual Plot
-50
0
50
-10 -5 0 5 10
X
Res
idu
als
ii bXaY ˆ
![Page 25: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/25.jpg)
2XY How does a Linear RegressionModel approximate (for X=-8,-7,…,7,8)
For these particular datathe regression
model finds
a = 24b = 0
r = 0
Y = X_squared Line Fit Plot
0
50
100
-10 -5 0 5 10
X
Y
Correlation is Zero: No LINEAR Relationship
Is there “no relationship” between X and Y?
There is an extremely strong (nonlinear) relationship here!
ii bXaY ˆ
![Page 26: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/26.jpg)
)ln(XY How does a Linear RegressionModel approximate (for X=1,2,…,15)
For these particular datathe regression
model finds
a = .54b = .16
The residuals have a systematic trend!!
This Linear Regressionis inappropriate!!
Y = ln(X) Line Fit Plot
0
2
4
0 5 10 15 20
X
Y
Y = ln(X) Residual Plot
-1
0
1
0 5 10 15 20
X
Res
idu
als
ii bXaY ˆ
![Page 27: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/27.jpg)
Correlation is not Causation!
Correlation between the size of your big toe and your performance on reading tasks is highly positive!
??
Lurking Third Variable: AGE
![Page 28: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/28.jpg)
Correlation is not Causation!
Only experimentationexperimentation allows us to attribute causationto the relationship between independent and
dependent variables.
![Page 29: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/29.jpg)
Ecological Correlation:Correlations between averages
are higher than correlations between individuals
X
Y
X Group averages
Y Group averages
![Page 30: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/30.jpg)
Problem of Restricted Range
GRE scores
Successin Graduate
School
Strong LinearRelationship
No LinearRelationship
![Page 31: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/31.jpg)
Extrapolations are Dangerous
Year
Number ofPassengers
![Page 32: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/32.jpg)
Regression toward the Mean
The term “Regression” is associated with Sir Francis Galton (1822 – 1911)
Picture taken from http://www.gene.ucl.ac.uk/
Galton (1885)“Regression towards Mediocrity
In Hereditary Stature”Journal of the Anthropological
Institute
![Page 33: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/33.jpg)
Regression toward the Mean
60. : and between n Correlatio
:son of IQ
:father of IQ
rYX
Y
X
Suppose:
XY ZZ 6.ˆ
![Page 34: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/34.jpg)
Regression toward Mediocrity??
60. : and between n Correlatio
:son of IQ
:father of IQ
rYX
Y
XXY ZZ 6.ˆ
2.1)0.2(6.Z :son mediocre morepredict willWe
0.2 Z:fathert intelligenVery
Y
X
2.1)0.2(6.Z :son dumb less apredict willWe
0.2 Z:father dumbVery
Y
X
Predictions are closer to zero (the mean) then the observations!!
![Page 35: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/35.jpg)
r=.60
2.0
1.2
2.0
1.2
XZ
YZ
![Page 36: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/36.jpg)
r=.60
2.0
1.2
XZ
YZ
Among families where the father is approximately 2 standard deviations above the mean, the average son is only about 1.2 standard deviations above the mean.
![Page 37: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/37.jpg)
Regression toward Mediocrity??
Do the sons just become more similar to each other than their fathers were?
![Page 38: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/38.jpg)
Regression toward Mediocrity??
: of Variance XZ
: of Variance YZ
1XZ
S
1YZ
S
Variability of the Z scores is the same!
No slide into mediocrity!!
![Page 39: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/39.jpg)
Regression toward the mean
When you have a lucky and exceptionally good performance in an exam,you expect to do worse next time, because there is no reason to believe
that you will be so exceptionally lucky again.
When you have a mental block and exceptionally bad performance in an exam,
you expect to do better next time, because there is no reason to believe that you will be so exceptionally unlucky again.
This does not mean that you are becomingmore and more average as time progresses.
It means that your average performance, as a reasonablepredictor for future performance, will lead to such a pattern
of relationships between observed and predicted performance
![Page 40: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/40.jpg)
Regression toward the mean
Your room mate makes a huge mess in your room. You complain. The next few days are cleaner.
Your room mate has cleaned up the room.You praise your room mate. The next few days the room gets dirtier.
Does this mean that punishment leads to better performance and reward leads to worse performance?
No….
![Page 41: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/41.jpg)
Regression toward the mean
Your room mate makes a huge mess in your room. You do nothing. The next few days are cleaner.
Your room mate has cleaned up the room.You do nothing. The next few days the room gets dirtier.
Your room mate simplymakes messes,
cleans them,makes messes,cleans them …
Your best guess for the future is an “average” level of messiness
![Page 42: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/42.jpg)
Implications for Research
It is very risky to study anything based on selection of extreme groups
Test RetestExtremes become less extreme
May look like a treatment effect!
![Page 43: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/43.jpg)
Relationships between Categorical Variables
Baby Held
Right-Handed Mother
Left-Handed Mother
Left 212 25
Right 43 7
237
50
255 32 287
Marginal Distributions
![Page 44: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/44.jpg)
Theory
“Mothers tend to hold their babies with the non-dominant hand,
so that the dominant hand is available to do stuff.”
![Page 45: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/45.jpg)
Relationships between Categorical Variables
Baby Held
Right-Handed Mother
Left-Handed Mother
Left
Right
.826 (82.6%)
.174 (17.4%)
.889(88.9%)
.111(11.1%)
Marginal Proportions (Percentages)
Vast majority of babies held leftVast majority of mothers right-handed
![Page 46: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/46.jpg)
Relationships between Categorical Variables
Baby Held
Right-Handed Mother
Left-Handed Mother
Left .894 .105
Right .860 .140
1 (100%)
1 (100%)
Conditional proportions,given side on which the baby is held
Absolute size not taken into account
![Page 47: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/47.jpg)
Relationships between Categorical Variables
Baby Held
Right-Handed Mother
Left-Handed Mother
Left .831 .781
Right .169 .219
1 (100%) 1 (100%)
Conditional proportions,given dexterity of mother
Absolute size not taken into account
![Page 48: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/48.jpg)
Relationships between Categorical Variables
1 (100%) 1 (100%)
For any given dexterity of the mother,there is an overwhelming tendency to hold the
baby on the left hand side.
Absolute size not taken into account
Baby Held
Right-Handed Mother
Left-Handed Mother
Left .831 .781
Right .169 .219
![Page 49: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/49.jpg)
Segmented Bargraphs
Segmented Bargraph
0
50
100
150
200
250
left holding right holding
Side Baby is held
Fre
qu
ency
left-handed
right-handed
![Page 50: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/50.jpg)
Segmented BargraphsSegmented Bargraph
0
50
100
150
200
250
300
right-handed left-handed
Dexterity
Fre
qu
ency
right holding
left holding
![Page 51: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/51.jpg)
Conclusion??
Lurking Third Variable?
Heart beat helps baby calm down
![Page 52: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/52.jpg)
Simpson’s Paradox
Admit Deny
Male 480 120
Female 180 20
Admit Deny
Male 10 90
Female 100 200
Business School
Law School
![Page 53: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/53.jpg)
Simpson’s Paradox
Admit Deny
Male 490 210
Female 280 220
Admit Deny
Male .70 30
Female .56 .44
Overall:
Overallconditional proportionsper gender
700
500
Men Priviliged!!Gender Discr.!!
![Page 54: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/54.jpg)
Simpson’s Paradox
Admit Deny
Male 480 120
Female 180 20
Admit Deny
Male 10 90
Female 100 200
Admit Deny
Male .80 .20
Female .90 .10
Admit Deny
Male .10 .90
Female .33 .67
600
200
100
300
WomenPriviliged!?!
WomenPriviliged!?!
![Page 55: Heads Up! Sept 22 – Oct 4 Probability Perceived by many as a difficult topic Get ready ahead of time](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f575503460f94c7c268/html5/thumbnails/55.jpg)
Simpson’s Paradox
Admit Deny
Male 480 120
Female 180 20
Admit Deny
Male 10 90
Female 100 200
Admit Deny
Male .80 .20
Female .90 .10
Admit Deny
Male .10 .90
Female .33 .67
600
200
100
300
However: Higher admission rate for male dominated discipline