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Ch. 22 2 - test

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Ch. 22 2 - test

Introduction to 2 - test

Structure of 2– test

Testing Stochastic Independence

3

1

2

Introduction to 2 - test

Structure of 2– test

Testing Stochastic Independence

Usage of - test

- Predicting whether Stock price index would be up or down:
There are only 2 categories

z – test

Sign test

- Predicting level of Stock price index by intervals:
There are categories more than 2

2– test

- If Average of cards in Box being only matter…

z – test

t - test

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

– test

Usage of - test

Usage of - test

- 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

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

z – test

t - test

- Testing the Null : aver. of box is 3.5

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

- test

An Ex. of - test

- Does a Gambler use a unfair die?

Result from 60 times casting

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

The Observed is much larger than the Expect.

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

=

(observed-expect)2

expect

- statistic

- 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.

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

Usage of - test

- 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 1,000개의 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

Degree of freedom of - test

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

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

- That distribution curves are right-tailed.
- As D.F. get larger, Shape of curve get more symmetric as moving to right.

D.F.

= 6-1 = 5

- distribution curve

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

Read the probability area in the first column of table.

p-value =

-statistics table : a section

14.2

면적과 자유도가 만나는 위치에 놓인 수치를 읽는다.

11.07

5% critical

value

15.09

1% critical

value

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

3

1

2

Introduction to 2 - test

Structure of 2– test

Testing on Stochastic Independence

Structure

Basic Data

Stochastic Model

A Frequency Table

In general,

Size of sample is

represented as

n

Ex) n=60

Box Model

Ex.) a Die Model:

A box containing

Cards having

numbers 1~6 on each

Random Sampling

with replacement from

a composition

Announced box

Recording frequencies

of each observation

And making the result

as a kind of table

1

(observed-expect)2

expect

Structure

Observed

Significance level (p-value)

2-statistics

Degree of Freedom

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

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%

0

3

1

2

Introduction to 2 - test

Structure of 2– test

Testing Stochastic Independence

It is by Real

It is by Chance

[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

Test for Stochastic Independence among variables

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

Gender and a Preferred hand (ratio)

Gender and a Preferred hand (frequency)

?

?

?

?

?

?

Right-handed Male

Right-handed Female

Left-handed Male

Left-handed Female

Ambidexter Male

Ambidexter Female

Designing a box model

- 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

Null vs Alternative

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

Calculate Expect values under the Null.

- -statistic

2 - test

When testing stochastic independence on a mn table, If there is no probability

restriction except stochastic independence, the D.F. will be (m-1)(n-1).

- Degree of Freedom

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

2 - test

- p-value

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

0.2%

p-value

12

- 자유도 2인 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.

Expected Frequencies

(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.