© asup -2006 probability and normal distribution 1 probability
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PROBABILITY
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INTRODUCTION TO PROBABILITY
We introduce the idea that research studies begin with a general question about an entire population, but actual research is conducted using a sample
POPULATION SAMPLEProbability
Inferential Statistics
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THE ROLE OF PROBABILITY IN INFERENTIAL STATISTICS
Probability is used to predict what kind of samples are likely to obtained from a population
Thus, probability establishes a connection between samples and populations
Inferential statistics rely on this connection when they use sample data as the basis for making conclusion about population
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PROBABILITY DEFINITION
The probability is defined as a fraction or a proportion of all the possible outcome divide by total number of possible outcomes
Probability of A =
Number of outcome classified as A
Total number of possible outcomes
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EXAMPLE If you are selecting a card from a
complete deck, there is 52 possible outcomes• The probability of selecting the king
of heart?• The probability of selecting an ace?• The probability of selecting red
spade? Tossing dice(s), coin(s) etc.
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PROBABILITY andTHE BINOMIAL DISTRIBUTION
When a variable is measured on a scale consisting of exactly two categories, the resulting data are called binomial (two names), referring to the two categories on the measurement
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PROBABILITY andTHE BINOMIAL DISTRIBUTION In binomial situations, the researcher
often knows the probabilities associated with each of the two categories
With a balanced coin, for examplep (head) = p (tails) = ½
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PROBABILITY andTHE BINOMIAL DISTRIBUTION
The question of interest is the number of times each category occurs in a series of trials or in a sample individual.
For example:• What is the probability of obtaining 15
head in 20 tosses of a balanced coin?• What is the probability of obtaining
more than 40 introverts in a sampling of 50 college freshmen
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TOSSING COIN Number of heads obtained in 2 tosses a
coin•p = p (heads) = ½•p = p (tails) = ½
We are looking at a sample of n = 2 tosses, and the variable of interest is X = the number of head
Number of heads in 2 coin tosses
The binomial distribution showing
the probability for the number of heads in 2
coin tosses0 1 2
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TOSSING COIN
Number of heads in 3 coin tosses
Number of heads in 4 coin tosses
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The BINOMIAL EQUATION
(p + q)n
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In an examination of 5 true-false problems, what is the probability to answer correct at least 4 items?
In an examination of 5 multiple choices problems with 4 options, what is the probability to answer correct at least 2 items?
LEARNING CHECK
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PROBABILITY and NORMAL DISTRIBUTION
In simpler terms, the normal distribution is symmetrical with a single mode in the middle. The frequency tapers off as you move farther
from the middle in either direction
μ
σ
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PROBABILITY and NORMAL DISTRIBUTION
Proportion below the curve B, C, and D area
μ X
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B and C area
X
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B and C area
X
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B, C, and D area
B + C = 1C + D = ½ B – D = ½
μ X
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B, C, and D area
B + C = 1C + D = ½ B – D = ½
μX
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The NORMAL DISTRIBUTION following a z-SCORE transformation
-2z -1z 0 +1z +2z
μ
34.13%
13.59%
2.28%
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-2z -1z 0 +1z +2z
μ = 166
34.13%
13.59%
2.28%
Assume that the population of Indonesian adult height forms a normal shaped with a mean of μ = 166 cm and σ = 7 cm
• p (X) > 180?
• p (X) < 159?
σ = 7
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-2z -1z 0 +1z +2zμ = 166
34.13%
13.59%
2.28%
Assume that the population of Indonesian adult height forms a normal shaped with a mean of μ = 166 cm and σ = 7 cm
• Separates the highest 10%?
• Separates the extreme 10% in the tail?
σ = 7
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-2z -1z 0 +1z +2z
μ = 166
34.13%
13.59%
2.28%
Assume that the population of Indonesian adult height forms a normal shaped with a mean of μ = 166 cm and σ = 7 cm
• p (X) 160 - 170?
• p (X) 170 - 175?
σ = 7
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EXERCISE
From Gravetter’s book page 193 number 2, 4, 6, 8, and 10