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Probability and Normal Distribution PROBABILITY © aSup -2006 1 Probability and Normal Distribution 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 Inferential Statistics SAMPLE Probability © aSup -2006 2 Probability and Normal Distribution 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 -2006 3 Probability and Normal Distribution 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 © aSup -2006 = Number of outcome classified as A Total number of possible outcomes 4 Probability and Normal Distribution 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 -2006 5 Probability and Normal Distribution 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 -2006 6 Probability and Normal Distribution 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 -2006 7 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 © aSup -2006 8 Probability and Normal Distribution 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 The binomial distribution showing the probability for the number of heads in 2 coin tosses © aSup -2006 0 1 2 Number of heads in 2 coin tosses 9 Probability and Normal Distribution TOSSING COIN Number of heads in 3 coin tosses Number of heads in 4 coin tosses © aSup -2006 10 Probability and Normal Distribution The BINOMIAL EQUATION (p + © aSup -2006 n q) 11 Probability and Normal Distribution LEARNING CHECK 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? © aSup -2006 12 Probability and Normal Distribution PROBABILITY and NORMAL DISTRIBUTION σ μ 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 © aSup -2006 13 Probability and Normal Distribution PROBABILITY and NORMAL DISTRIBUTION μ X Proportion below the curve B, C, and D area © aSup -2006 14 Probability and Normal Distribution B and C area X © aSup -2006 15 Probability and Normal Distribution B and C area X © aSup -2006 16 Probability and Normal Distribution B, C, and D area μ X B+C=1 C+D=½B–D=½ © aSup -2006 17 Probability and Normal Distribution B, C, and D area X μ B+C=1 C+D=½B–D=½ © aSup -2006 18 Probability and Normal Distribution The NORMAL DISTRIBUTION following a z-SCORE transformation 34.13% 13.59% 2.28% -2z -1z 0 +1z +2z μ © aSup -2006 19 Probability and Normal Distribution 34.13% σ=7 13.59% 2.28% -2z -1z 0 μ = 166 +1z +2z 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? © aSup -2006 20 Probability and Normal Distribution 34.13% σ=7 13.59% 2.28% -2z -1z 0 μ = 166 +1z +2z 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? © aSup -2006 21 Probability and Normal Distribution 34.13% σ=7 13.59% 2.28% -2z -1z 0 +1z μ = 166 +2z 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? © aSup -2006 22 Probability and Normal Distribution EXERCISE From Gravetter’s book page 193 number 2, 4, 6, 8, and 10 © aSup -2006 23