05. probability and probability distributions

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Probability and Probability Distributions

What is Probability ? If we toss a fair coin, there are two possible outcomes, a head (H) or a tail (T); that is N=2. So the P(H) = (0.5). If a toss of two coins, how many possible outcomes ?

First coin

Second coin H T

H

T

H T

Possible outcomes : HH, HT, TH, TT

What is the probability of : - P(2H) - P (at least 1H) -P( 1H and 1T)

A pregnant woman wonders about the chance of having a boy or a girl baby. Vital statistics datas indicate there are about 1056 live births of boys for every 1000 live birth of girls, so she estimate her probability of having a boy as 1056 / 2056 = 0.514

Probability : Probability applies exclusively to a future event, never to a past (outcome of event is unknown) Probability may be used to measure the uncertainty of the outcome of such events, for example , the probability of surviving to age 80 of developing cancer Probability statements are Numeric, in the range of 0 1 ( 0: the event will not happen, 1 the event will happen with certainty)

Events : Mutually exclusive event Not mutually exclusive event Complementary event Independence event

Mutually exclusive event : Event that cannot happen simultaneously, that is, if one event happen, the other event cannot happen For example : toss a coin, baby born (boy or girl) P(A or B) = P(A) + P(B) P (A U B)..P(A union B)

Not mutually exclusive event : Two event can happen simultaneously, so a part of two events are intersection For example : the event A that a 30 year old woman lives to see her 70th birthday and the event B that 30 years old husband is still alive at age 70. A and B would be the event that both the 30 year-old woman and her husband are alive at age 70.

Not mutually exclusive event : P(A or B) = P(A) + P(B) P(AB)

Event A

AB Event B

P(both A and B) = P(A)xP(B)

Complementary event Event Ac is the complement of event A

A

Ac

For example : 100 patients Lung cancer, 20 patients still a live for 3 years.P(A) = 20/100 = 0.2), so that 80 patients Acdied for 3 year 0.8)1 P(A) )=80/100 = P(

Independent event Two events are independent if the occurrence of one has no effect on the chance of occurrence of the other. The multiplication rule For example : the outcomes of repeated tosses of a coin, because the outcomes of one toss does not affect the outcomes of any future toss.

Probability Distributions A key application of probability to statistics is estimating the probabilities that are associated with the occurrence of different events. Help us reach a decision whether certaint events are significant or not Mathematical distributions

Probability Distributions Variables continue Normal Distributions Sample Mean Distributions ( t- student distributions) F Distributions

Variables discrete : Binomial distributions Chi Square distributions

the Normal Distribution Properties : Symmetrical Bell shaped curve extending infinitely in both directions Have area under the curve. The total is 1 It is a theoretical distribution defined by two parameters: and The mean and median of a Normal Distribution are equal(Kuzma, 2005; p.

Area under the Normal Curve99,74% 95,45% 68,26%

200 800 -3 +3

300 -2

400 -1

500

600

700

+1 +2

(Kuzma, 2005; p.

DATA YG MEMILIKI DISTRIBUSI MENDEKATI DISTRIBUSI NORMAL Sebagian besar data yg berasal dari hasil pengukuran variabel kuantitatif ( interval , ratio ); Sampling distribution of any quantitative data with n 30; Sampling distribution of data from dummy variables (yes or no, 0 or 1, etc) in which p x n 5.00. Examples n = 20, p = 0.25 (25% alkoholik)

The Standard Normal Distribution There are infinitely many Normal Distributions depending on the values of and . The Standard Normal Distribution is a particular Normal Distribution for which probabilities have been tabulated = 0 and = 1. (Petry, 2000; p. 21)

Area under the Normal CurveNormal Distribution of x with mean and variance 2

(a)

68%

-2,58 -1,96 - 95% 99%

+ +1,96 +2,58

68%

Standard Normal Distribution of Z

(b)

-2,58 -1,96 +2,58

-1

0

+1 +1,96

(Petry, 2000; p. 21)

the Normal Curve A standardized unit used is the standardized score, Z If a variable is normally distributed, then any individual raw score can be converted into a corresponding Z score (Z value/ standard x Score) Z= Formula :

(Kuzma, 2005; p.

Example 1 What is the proportion of persons having Stat score between 500 and 650 ? X 500, sd=100Z= Z= x 500 500 = =0 100

x 650 500 = = 1,5 1000.5 0.4332 500 650 0

Z table is 0.4332

SAT math Z

1,5

= 120 mmHg = 10 mmHgProbabilitas < 140 mm Hg ?

120

140

Z = ( 140 120 ) / 10 = 2.0BP < 140 = ?

Lihat Tabel Z = 0,0228

Berapa peluang untuk mendapatkan Tekanan darah antara 100 dan 130 mm Hg ?

0.0228 100 Z = -2.0 Tabel : 0,0228 130

0,1587

Z = 1.0 0.1587

BP antara 100 dan 130 = 1 ( 0.0228 + 0.1587 ) = 0,8185 ( 81,85 % )

example 2 :We assume that the systolic blood pressure of large group Of adult men are approximately normally distributed, That the mean systolic BP is 120 mmHg, and standard deviation is 10 mmHg. If one adult men is randomly selected What is the probabilty that his systolic BP will be less than 140 mmHg

The t-distribution Derived by W.S Gossett who published under the pseudonym Student, it is often called Students tdistribution Using Degree of Freedom Its shape similar to that of the Standard Normal Distribution, but it is more spread out longer tails. Its shape approaches Normality as the degrees of freedom increase It is particularly useful for calculating confidence intervals for and testing hypotesis about one or two means We use it when the population SD is not known

HIMPUNAN NILAI x SAMPEL (distribusi sampel )POPULASI n1

x1n2 n3

x2 x3 xiGambar grafik himpunan nilai x ini mempunyai gambar spt distribusi normal

dstCiri-ciri distribusi normal dapat diterapkan utk menduga

Comparison of t Distribution and the Normal Distribution

Probability 0.3

Normal Distribution t (df = 20) t (df = 2)

0.2

0.1

0

-4 4

-2 t

0

2 (Kuzma, 2005; p.

Degree of Freedom The sample size minus the number of parameters that have to be estimated to calculate the statistic. They indicate the extent to which the observation are free to vary

Distribusi Binomial Ciri ciri ; Bila jumlah n tetap dan p kecil maka distribusi yang dihasilkan akan menceng kanan dan bila p makin besar maka kemiringan akan berkurang dan bila p = 0.5 maka distribusi akan simetris dan bila p > 0.5 maka distribusi yang dihasilkan menceng kiri Bila p tetap dengan jumlah n yang makin besar maka akan dihasilkan distribusi yang mendekati distribusi simetris

Syarat distribusi Binomial1. Tiap peristiwa mempunyai 2 hasil 2. Probabilitas dari setiap peristiwa harus selalu tetap 3. Event yang dihasilkan bersifat independen X = np sd = pq / n

The Chi Square Distribution It is right skewed distribution taking positive value It is characterized by its degree of freedom Its shape depends on the degrees of freedom. It become more symmetrical and approaches Normality as they increase It is particularly useful for analising categorical data

The Chi Square Distribution for Varying Degrees of Freedom

Probability 0.3 1 df 4 df 0.2 6 df 0.1 X2 value

0

5 3,84

10 9,49

15 12,59

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