Midterm Review
Econ 240A
1
The Big Picture
2
The Classical Statistical Trail
Rates &
Proportions
Inferential
Statistics
Descriptive
Statistics
Probability
Discrete Random
Application
Binomial
Variables
Power 4-#4
Discrete Probability Distributions; Moments
Descriptive Statistics
Power One-Lab One
Concepts
central tendency: mode, median, mean
dispersion: range, inter-quartile range,
standard deviation (variance)
Are central tendency and dispersion
enough descriptors?
4
Concepts
Normal Distribution
– Central tendency: mean or average
– Dispersion: standard deviation
Non-normal distributions
Density Function for the Standardized Normal Variate
Draw a Histogram
0.45
80
0.4
Frequency
0.35
Density
0.3
0.25
0.2
60
40
20
0.15
0
0.1
15
0.05
-4
-3
-2
-1
45
60
75
90 105 120
Bills
0
-5
30
0
1
2
3
4
5
Standard Deviations
5
The Classical Statistical Trail
Rates &
Proportions
Descriptive
Statistics
Inferential
Statistics
Classicall
Application
Modern
Probability
Discrete Random
Binomial
Variables
Power 4-#4
Discrete Probability Distributions; Moments
Exploratory Data Analysis
Stem and Leaf Diagrams
Box and Whiskers Plots
7
Weight Data
Males:
140 145 160 190 155 165 150 190 195 138
160 155 153 145 170 175 175 170 180 135 170 157
130 185 190 155 170 155 215 150 145 155 155 150
155 150 180 160 135 160 130 155 150 148 155 150
140 180 190 145 150 164 140 142 136 123 155
Females:
140 120 130 138 121 125 116 145 150 112 125 130
120 130 131 120 118 125 135 125 118 122 115 102
8
115 150 110 116 108 95 125 133 110 150 108
9
Box Diagram
median
First or lowest quartile;
25% of observations below
Upper or highest quartile
25% of observations above
10
3rd Quartile + 1.5* IQR = 156 + 46.5 = 202.5; 1st value below =195
11
The Classical Statistical Trail
Rates &
Proportions
Inferential
Statistics
Descriptive
Statistics
Probability
Discrete Random
Application
Binomial
Variables
Power 4-#4
Discrete Probability Distributions; Moments
Power Three - Lab Two
Probability
13
Operations on events
The event A and the event B both
occur: ( A  B)
Either the event A or the event B
( A  B)
occurs or both do:
The event A does not occur, i.e.not
A:
A
14
Probability statements
Probability of either event A or event B
p( A  B)  p( A)  p( B)  p( A  B)
– if the events are mutually exclusive, then
p( A  B)  0
probability of event B
p( B)  1  p ( B )
15
Conditional Probability
Example: in rolling two dice, what is
the probability of getting a red one
given that you rolled a white one?
– P(R1/W1) ?
16
In rolling two dice, what is the probability of getting a red one given
that you rolled a white one?
17
Conditional Probability
Example: in rolling two dice, what is
the probability of getting a red one
given that you rolled a white one?
– P(R1/W1) ?
p( R1 / W 1)  p( R1  W 1) / p(W 1)  (1 / 36) /(1 / 6)
18
Independence of two events
p(A/B) = p(A)
– i.e. if event A is not conditional on event
B
– then p A  B  p( A) * p( B)
19
The Classical Statistical Trail
Rates &
Proportions
Inferential
Statistics
Descriptive
Statistics
Probability
Discrete Random
Application
Binomial
Variables
Power 4-#4
Discrete Probability Distributions; Moments
Power 4 – Lab Two
21
Three flips of a coin; 8 elementary outcomes
H
p
p
p
H
1-p
H
H
T
p
1-p
T
T
H
H
T
1-p
p
T
H
1-p
T
T
3 heads
2 heads
2 heads
1 head
2 heads
1 head
1 head
0 heads
22
The Probability of Getting k Heads
The probability of getting k heads
(along a given branch) in n trials is:
pk *(1-p)n-k
The number of branches with k heads
in n trials is given by Cn(k)
So the probability of k heads in n trials
is Prob(k) = Cn(k) pk *(1-p)n-k
This is the discrete binomial distribution
where k can only take on discrete
values of 0, 1, …k
23
Expected Value of a discrete
random variable
E(x) =
n
 x(i) * p[ x(i)]
i 0
the expected value of a discrete
random variable is the weighted
average of the observations where
the weight is the frequency of that
observation
24
Variance of a discrete random
variable
VAR(xi) =
n
2
{[
x
(
i
)

E
[
x
(
i
)]
}
p[ x(i )]

i 0
the variance of a discrete random
variable is the weighted sum of each
observation minus its expected
value, squared,where the weight is
the frequency of that observation
25
Lab Two
The Binomial Distribution, Numbers & Plots
– Coin flips: one, two, …ten
– Die Throws: one, ten ,twenty
The Normal Approximation to the Binomial
– As n
∞, p(k)
N[np, np(1-p)]
– Sample fraction of successes:
ˆp  k / n, E ( pˆ )  np / n  p,Var ( pˆ )  np(1  p) / n 2
pˆ ~ N [ p, p(1  p) / n]
26
Lab Three and Power 5,6
1/ 2[( z 0) /1]2
f ( z)  [1/ 2 ] * e
Z~N(0,1)
Prob(1.96≤z≤1.96)=0.95
ˆ
ˆ
ˆ
Density Function for the Standardized Normal Variate
0.45
0.4
0.35
Density
0.3
0.25
0.2
2.5%
0.15
z  [ p  E ( p )] /  pˆ
2.5%
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
3
-1.96Standard Deviations 1.96
4
5
ˆ pˆ  pˆ (1  pˆ ) / n
prob (1.96  [ pˆ  E ( pˆ )] / ˆ pˆ  1.96)  0.95
prob (1.96 *ˆ pˆ  pˆ  p  1.96 *ˆ pˆ )  0.95
prob (1.96 * ˆ pˆ  p  pˆ  1.96 *ˆ pˆ )  0.95
prob ( pˆ  1.96 *ˆ pˆ  p  pˆ  1.96 *ˆ pˆ )  0.95
27
Hypothesis Testing: Rates & Proportions
One-tailed test:
Step #1:hypotheses
H0 : p  f
Ha : p  f
One-tailed test:
Step #2: test statistic
z  [ pˆ  E ( pˆ )] / ˆ pˆ  [ pˆ  f ] / ˆ pˆ ,
ˆ pˆ  pˆ (1  pˆ ) / n
Density Function for the Standardized Normal Variate
One-tailed test:
Step #3: choose 
e.g.  = 5%
0.4
Z=1.645
0.35
0.3
Density
Step # 4: this determines
The rejection region for H0
0.45
Reject if
0.25
( pˆ  f ) / ˆ  1.645
0.2
0.15
5%
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
Standard Deviations
2
3
4
28
5
Remaining Topics
Interval estimation and hypothesis
testing for population means, using
sample means
Decision theory
Regression
– Estimators
OLS
Maximum lilelihood
Method of moments
– ANOVA
29
Midterm Review Cont.
Econ 240A
30
Last Time
31
The Classical Statistical Trail
Rates &
Proportions
Inferential
Statistics
Descriptive
Statistics
Probability
Discrete Random
Application
Binomial
Variables
Power 4-#4
Discrete Probability Distributions; Moments
Remaining Topics
Interval estimation and hypothesis
testing for population means, using
sample means
Decision theory
Regression
– Estimators
OLS
Maximum lilelihood
Method of moments
– ANOVA
33
Lab Three
Power 7
Population
Random variable x
Distribution f(m, 2
f?
Pop.
Sample
Sample Statistic:
x ~ N (m , )
2
Sample Statistic
n
s 2   ( xi  x ) 2 /( n  1)
i 1
34
f(x) in this example is Uniform
X~U(0.5, 1/12)
E(x) = 0.5
Var(x) = 1/12
f(x)
0
x
Nonetheless, from the central
Limit theorem, the sample mean
Has a normal density
1
x ~ N [0.5, (1 / 12) / n ]
Density Function for the Standardized Normal Variate
0.45
z  [ x  E ( x )] /  x
0.4
0.35
z  [ x  0.5] / (1 / 12) / n
Density
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
Standard Deviations
2
3
4
5
35
Histogram of 50 Sample Means, Uniform, U(0.5, 1/12)
0.
95
M
or
e
0.
85
0.
75
0.
65
0.
55
0.
45
0.
35
0.
25
0.
15
20
15
10
5
0
0.
05
Frequency
Histogram of 50 sample means
Sample Mean
Average of the 50 sample means: 0.4963
36
Inference
ingeneral , x ~ f ( m ,  2 )
fromthecentra lim ittheoremweknow
x ~ N [ m ,  2 / n]
Density Function for the Standardized Normal Variate
so, z  [ x  m ] /( / n )
0.45
0.4
Pr ob(1.96  z  1.96)  0.95
Pr ob(1.96( / n )  ( x  m )  1.96( / n ))  0.95
Z=-1.96
Density
Pr ob(1.96  ( x  m ) /( / n )  1.96)  0.95
0.35
0.3
0.25
Z=1.96
0.2
2.5%
Pr ob(1.96( / n )  ( m  x )  1.96( / n ))  0.95
0.15
2.5%
0.1
0.05
Pr ob( x  1.96( / n )  ( m )  x  1.96( / n ))  0.95
0
-5
-4
-3
-2
-1
0
1
2
3
Standard Deviations
37
4
5
Confidence Intervals
If the population variance is known, use the
normal distribution and z
z  ( x  m ) /( / n )
If the population variance is unknown, use
Student’s t-distribution and t
t  ( x  m ) /( s / n )
where, s 
i n
2
(
x

x
)
/( n  1)
 i
i 1
38
t-distribution
Text p.253
Normal
compared to t
t distribution
as smple size
grows
39
Appendix B
Table 4
p. B-9
40
Hypothesis
tests
Step two: choose the test statistic
Step One: state the
hypotheses
H0 : m  v
You choose v
Ha : m  v
2-tailed test
Step Three: choose the size
Of the Type I error, =0.05
Density Function for the Standardized Normal Variate
z  [ x  E ( x )] /  x
z  [ x  v] /( / n )
Step four: reject the null hypothesis
if the test statistic is in the
Rejection region
0.45
0.4
Density
Z=-1.96
0.35
Z=1.96
0.3
0.25
0.2
2.5%
2.5%
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Standard Deviations
41
True State of Nature
p = 0.5
Accept null
Decision
Reject null
No Error
1-
Type I error 
C(I)
P > 0.5
Type II error b
C(II)
No Error
1-b
E[C] = C(I)*  + C(II)* b
42
Regression Estimators
Minimize the sum of squared
residuals
Maximum likelhood of the sample
Method of moments
43
Minimize the sum of squared residuals
n
n
Min. eˆ   ( yi  yˆ i )
i 1
2
i
2
i 1
n
2
ˆ
  ( yi  aˆ  bxi )
i 1
n
  eˆ / aˆ  0, y  aˆ  bˆx
i 1
2
i
44
Maximum likelihood
n
2
2
ˆ
ˆ
 ln Lik /   0,   [ ei ] / n
2
i 1
Method of moments
n
n
n
bˆ   ( yi  y )( xi  x ) / n( xi  x ) 2
2
ˆ
b   ( yi  y )( xi  x ) /  ( xi  x )
i 1
i 1
i 1
i 1
45
Inference in Regression
Interval estimation
t  [bˆ  Ebˆ] / ˆ bˆ  [bˆ  b] / ˆ bˆ
prob(t.025  [bˆ  b] / ˆ bˆ  t.975 )  0.95
prob(ˆ bˆt.025  bˆ  b  ˆ bˆt.975 )  0.95
prob(ˆ bˆ t.025  b  bˆ  ˆ bˆt.975 )  0.95
prob(bˆ  ˆ bˆ t.025  b  bˆ  ˆ bˆt.975 )  0.95
46
Estimated Coefficients, Power 8
taˆ  [aˆ  E (aˆ )] / ˆ aˆ  (1.204  0) / 0.727  1.41
Coefficients
Standard
Error
t Stat
P-value
Lower 95%
Upper
95%
-1.02377776
0.727626534
-1.40701
0.167999648
-2.499472762
0.451917
0.06565026
0.001086328
60.43316
8.58311E-38
0.063447085
0.067853
â
Intercept
b̂
X Variable 1
47
Appendix B
Table 4
p. B-9
48
Inference in Regression
Hypothesis testing
Step One
State the hypothesis
H0 : b  0
Ha : b  0
Step Three
Choose the size of the
Type I error, 
Step Two
Choose the test statistic
t  [bˆ  b] / ˆ bˆ
Step Four
Reject the null hypothesis if the
Test statistic is in the rejection region
49