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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