© aSup Probability and Normal Distribution  1 PROBABILITY

© aSup Probability and Normal Distribution  2 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 POPULATIONSAMPLE Probability Inferential Statistics

© aSup Probability and Normal Distribution  3 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

© aSup Probability and Normal Distribution  4 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

© aSup Probability and Normal Distribution  5 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.

© aSup Probability and Normal Distribution  6 PROBABILITY and THE 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

© aSup Probability and Normal Distribution  7 PROBABILITY and THE BINOMIAL DISTRIBUTION  In binomial situations, the researcher often knows the probabilities associated with each of the two categories  With a balanced coin, for example p (head) = p (tails) = ½

© aSup Probability and Normal Distribution  PROBABILITY and THE 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 8

© aSup Probability and Normal Distribution  9 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 tosses 0 1 2

© aSup Probability and Normal Distribution  10 TOSSING COIN Number of heads in 3 coin tosses Number of heads in 4 coin tosses

© aSup Probability and Normal Distribution  11 The BINOMIAL EQUATION (p + q) n

© aSup Probability and Normal Distribution  12  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

© aSup Probability and Normal Distribution  13 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

© aSup Probability and Normal Distribution  14 PROBABILITY and NORMAL DISTRIBUTION Proportion below the curve  B, C, and D area μ X

© aSup Probability and Normal Distribution  15 B and C area X

© aSup Probability and Normal Distribution  16 B and C area X

© aSup Probability and Normal Distribution  17 B, C, and D area B + C = 1 C + D = ½  B – D = ½ μ X

© aSup Probability and Normal Distribution  18 B, C, and D area B + C = 1 C + D = ½  B – D = ½ μ X

© aSup Probability and Normal Distribution  19 The NORMAL DISTRIBUTION following a z-SCORE transformation -2z -1z 0 +1z +2z μ 34.13% 13.59% 2.28%

© aSup Probability and Normal Distribution  20 -2z -1z 0 +1z +2z μ = % 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

© aSup Probability and Normal Distribution  21 -2z -1z 0 +1z +2z μ = % 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

© aSup Probability and Normal Distribution  22 -2z -1z 0 +1z +2z μ = % 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) ? p (X) ? σ = 7

© aSup Probability and Normal Distribution  23 EXERCISE  From Gravetter’s book page 193 number 2, 4, 6, 8, and 10