1. SPECIAL PROBABILITY DISTRIBUTION
Budiyono
2011

2. BINOMIAL DISTRIBUTION (Bernoulli Distribution)

3. Solution:

4. NORMAL DISTRIBUTION (Gaussian Distribution)

5. Normal Distribution Curve
symetri axes
area = 1 • • • • • • •
x=µ
x=µ-σ
x=µ+σ
x=µ+2σ
x=µ-2σ
x=µ+3σ
x=µ-3σ

6. STANDARD NORMAL DISTRIBUTION N(0,1)

7. STANDARD NORMAL DISTRIBUTION N(0,1)

8. STANDARD NORMAL DISTRIBUTION N(0,1)
area = 1 • • • • • • • -3 -2 -1
z=0 1 2 3

9. Standard Normal Distribution Table
This area can be found by using a standard normal distribution tablel z
This area can be thought as a probability appearing Z between 0 and z, written as P(Z|0<Z<z)

10. Example using a standard normal distribution table
Area = ?
0.4115
P(Z|0<Z<1.35) =
1.35
P(Z|Z>1.35) = =
0.05
.4115
1.3

11. Example using a standard normal distribution table
Area =?
-1.24
0.98
Area = =
0.7290

12. On a group of 1000 students, the mean of their score is 70
On a group of 1000 students, the mean of their score is 70.0 and the standard deviation is 5.0. Assuming that the score are normally distributed. How many students have score between 73.6 dan 81.9?
Problem
Solution
µ = 70.0;
σ = 5.0;
X1 = 73.6;
X2 = 81.9;
We transform X into z by using the formulae:

13. Area = – =
0.72
2.38
P(73.6<X<81.9) = P(Z|0.72<Z<2.38) =
So, the number of students having score between 73.6 and 81.9 is x 1000 = 227 student

14. STANDARD NORMAL DISTRIBUTION N(0,1)
sumbu simetri
luas = 1 • • • • • • • -3 -2 -1
z=0 1 2 3

15. STANDARD NORMAL DISTRIBUTION N(0,1)
0.3413
0.4772
0.0013
0.4987 • • • • • • • -3 -2 -1
z=0 1 2 3
z0.0013
z0.0228
z0.8413
z0.5000
z0.1587

16. Critical Value and Crtitical Region on N(0,1)
Significance level, usually denoted by α •
It is called critical region (daerah kritis), denoted by CR
It is called critical value (nilai kritis) (CV), denoted by zα
CR = {z | z > zα}

17. Getting zα for α = 25% • zα z0.25 = ? 0.67 .07 0.6 α = 25% 0.25 0.25
0.2486
0.2500

18. Getting zα for α = 10% • zα z0.10 = ? 1.28 .08 1.2 α = 10% 0.40 0.10
0.3997
0.4000

19. Getting zα for α = 5% • zα z0.05 = ? 1.645 .04 .05 1.6 α = 5% 0.45
0.4495
0.4500
0.4505

20. The Important Values zα• zα
Z0.025 = 1.96
Z0.01 = 2.33
Z0.005 = 2.575
Z0.05 = 1.645

21. Properties of zα α α • •
z1-α zα
z1-α =
-zα

22. STUDENT’S t DISTRIBUTION

23. Critical Values for t distributionα
Seen from the table •
tα ; Ʋ
t0.10 ; 12 =
1.356
t0.05 ; 12 =
1.782
t0.005 ; 28 =
2.763
t0.01 ; 24 =
2.492

24. Properties of tα;n α α • •
t1-α; n
tα ; n
t1-α; n =
-tα; n

25. THE CHI-SQUARE DISTRIBUTION

26. Critical Value for Chi-Square Distributionα α
Seen from the table • •
Properties:
Example
48.278
11.070

27. THE F DISTRIBUTION

28. Critical Values for F distributionα α
Seen from the table • •
Properties:
Examples:
3.29
26.87

29. Critical Values for F distribution
0.05 •
F0.95; 2, 15
F0.95; 2, 15 =
= 0.051