Ch. 22 2 - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics


Statistics for EconomistStatistics for Economist

Ch. 22 Ch. 22 2 2 - test- test

1.1. Introduction to Introduction to 22 - test - test

2.2. Structure of Structure of 22 – test – test

3.3. Testing Stochastic Testing Stochastic IndependenceIndependence

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STATISTISTATISTICSCS

INDEX

3Testing Stochastic Independence

1Introduction to 2 -

test

2 Structure of 2 – test

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1. Introduction to 2 - test

Usage of - test

Usage of - test

2

Predicting whether Stock price index would be up or down:

There are only 2 categories z – testSign test

Predicting level of Stock price index by intervals:

There are categories more than 2 2– test

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If Average of cards in Box being only matter…

z – testt - test

If the number of several kinds of cards in box being matter…

– test2


Usage of - test

Usage of - test

2

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- test

Testing the Null : aver. of box is 3.5 z – testt - test

Testing the Null : the prob. one card drawn out is 1/6 each

Drawing out Cards having numbers from 1 to 6 on each other from a box with replacement

2

2 -test indicates whether we can consider observed sample as from random sampling when we know about composition of contents in box

z-test or t-test indicate whether we can consider observed sample as from random sampling when we only know average of box

2 -test indicates whether we can consider observed sample as from random sampling when we know about composition of contents in box

z-test or t-test indicate whether we can consider observed sample as from random sampling when we only know average of box


Usage of - test

Usage of - test

2

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Does a Gambler use a unfair die?

4 3 3 1 2 3 4 6 5 6

2 4 1 3 3 5 3 4 3 4

3 3 4 5 4 5 6 4 5 1

6 4 4 2 3 3 2 4 4 5

6 3 6 2 4 6 4 6 3 2

5 4 6 3 3 3 5 3 1 4

Result from 60 times casting

Number Observed Expect

1 4 10

2 6 10

3 17 10

4 16 10

5 8 10

6 9 10

합 60 60

Result from 60 times drawing out cards having numbers from 1 to 6 on each with replacement from a box

The Observed is much larger than the Expect.


An Ex. of - test

An Ex. of - test

2

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- statistic - statistic2

Only one or two ridiculous columns can not determine whether whole data’s ridiculousness.

There needs certain indicators presenting overall difference between the observed and the expect getting all information together.

= = (observed-expect)2

expect

2The bigger -statistic means there is big difference between Observed values and Expect values.

2

2.1410

)109(

10

)108(

10

)1016(

10

)1017(

10

)106(

10

)104( 2222222


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The earned value, 14.2 is too big to think the model is true.

It may be possible to earn such a large number when casting a fair die in 60 times, but the size of possibility matters.

Earn 1,000 of 2- statistics by 1,000 times repetition of casting a fair die 60 times and then calculating the 2- statistic.

When applying 2- statistics to a histogram (in fact, a Empirical Histogram of 2-distribution), the Area of histogram right to the value 14.2.

The ratio of 1,000 개의 2-statistics to 1,000 statistics more than 14.2

The 2- statistics more than 14.2 are strong evidences against the model.

How big the probability would be that One stochastic model produce such a strong contrary evidence against itself ? Meaning of p-value

The 2- statistics more than 14.2 are strong evidences against the model.

How big the probability would be that One stochastic model produce such a strong contrary evidence against itself ? Meaning of p-value


Usage of - test

Usage of - test

2

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Degree of freedom of - test

Degree of freedom of - test

2

0

10

20

0 5 10 15 20 25 30

%

5자유도

10자유도

2 –distribution curve responding to D.F.(5) and D.F.(10)

That distribution curves are right-tailed.

As D.F. get larger, Shape of curve get more symmetric as moving to right.

That distribution curves are right-tailed.

As D.F. get larger, Shape of curve get more symmetric as moving to right.

As Model is designed in the concrete, It is meaningless to infer the population parameter : D.F. = the number of terms used in calculating 2 -statistic - 1

As Model is designed in the concrete, It is meaningless to infer the population parameter : D.F. = the number of terms used in calculating 2 -statistic - 1

D.F.

= 6-1 = 5


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- distribution curve

- distribution curve

2

-statistics table : a section

2

자유도 50% 10% 5% 1%

3 2.37 6.25 7.8211.34

4 3.36 7.78 9.4913.28

5 4.35 9.2411.07

15.09

6 5.3510.65

12.59

16.80

7 6.3512.02

14.07

18.48

The size of area right to 14.2 is the value between 5% and 1%

2-distribution curve in D.F.(5)

Read the probability area in the first column of table.

14.2

p-value =

면적과 자유도가 만나는 위치에 놓인 수치를 읽는다 .

11.075% critical

value

15.091% critical

value


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INDEX

3Testing on Stochastic Independence


test


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2. Structure of 2 – test

StructureStructure

In general, Size of sample is represented as

nEx) n=60

Box ModelEx.) a Die Model:A box containing

Cards having numbers 1~6 on each

Random Sampling with replacement from

a composition Announced box

Basic Data Stochastic Model

Recording frequencies of each observation

And making the result as a kind of table

A Frequency Table

11

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StructureStructure

In the case of no need to infer the population

parameter, D.F. is as below

the number of terms used in calculating 2-statistic - 1

Ex) 6-1=5

2-statistics Degree of Freedom

The p-value is the size of area right to

2- statistic under the 2-distribution curve of

corresponding D.F.

Ex) p-value=1.4%

Observed Significance level (p-value)

(observed-expect)2expect

00

2. Structure of 2 – test

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INDEX

3Testing Stochastic Independence


test


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3.Testing Stochastic Independence

Test for Stochastic Independence among variables

Test for Stochastic Independence among variables

Is it stochastic independent? : Left-handedness and Gender?

Male Female

Right 934 1,070

Left 113 92

Ambidexter 20 8

M(100%) F(100%)

Right 87.5% 91.4%

Left 10.6% 7.9%

Ambidexter 1.9% 0.7%

Gender and a Preferred hand (frequency)

Gender and a Preferred hand (ratio)

[Physiology] As Women’s left brain is more activated than Men’s, More Right-handedness.[Sociology] Women got forced more to use Right hand than men.

The Ratio of preferred hand is Identical to both Men and Women, Difference above is just by chance

It is by ChanceIt is by Real

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Designing a box model

Designing a box model

? Right-handed Male ?Right-handed Female

? Left-handed Male ? Left-handed Female

? Ambidexter Male ? Ambidexter Female

Make a Box model under the assumption that 2,237 people of sample are randomly drawn out from population.

2,237 times of Random Sampling without replacement

Male Female

Right 934 1,070

Left 113 92

Ambidexter 20 8

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Null vs AlternativeNull vs Alternative

Observed Frequency Expected Frequency

Male Female Male Female

Right 934 1,070 956 1.048

Left 113 92 98 107

Ambidexter 20 8 13 15

Gender and a Preferred hand

Difference in ratio between Gender and a

Preferred hand

Null Mutually IndependentJust a coincidence

occurred during sampling process

Alternative

A practical relation exists

Reflects practical difference of population

Calculate Expect values under the Null.

Observed and Expect per each category (Calculation of Expect will be following)

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2 - test2 - test

1215

)158(

13

)1320(

107

)10792(

98

)98113(

048,1

)048,1070,1(

956

)956934( 2222222

-statistic2

Degree of Freedom

Male Female Sum

Right -22 22 0

Left 15 -15 0

Ambidexter 7 -7 0

Sum 0 0 0

Difference between Observed and Expect per each category As two values are given, the

rests will be determined automatically :

Only two deviations are free among 6

D.F. = (3-1)(2-1) = 2

When testing stochastic independence on a m n table, If there is no probabilityrestriction except stochastic independence, the D.F. will be (m-1) (n-1).

When testing stochastic independence on a m n table, If there is no probabilityrestriction except stochastic independence, the D.F. will be (m-1) (n-1).

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2

p-value

2-distribution curve of D.F.(2)

12

0.2%

p-value

자유도 2 인 In 2-distribution curve of D.F.(2), Size of the area right to 12 is 0.2%. So. Reject the Null.

We can tell Gender and a preferred hand : mutually dependent.

2 - test2 - test

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Expected Frequencies


Observed FrequenciesRatio


Male Female Male Female

Right 934 1,070 89.6% 956 1,048

Left 113 92 9.1% 98 107

Ambidexter 20 8 1.3% 13 15

Sum 1,067 1,170 100% 1,067 1,170

(934+1,070)/2,237 89.6% :

If gender and a preferred hand were mutually independent, Number of right-handed male is expected to be 956 (89.6% of the 1,067 male) Getting the Expect using both Sample data and Null hypothesis. As Getting the expect by inference, this results in reduction of D.F.

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Ch. 22 2 - test