1. PROBABILITY

2. OVERVIEW Relationships between samples and populations most often are described in terms of probability. Relationships between samples and populations most often are described in terms of probability.

3. INTRODUCTION TO PROBABLITY DEFINITION : DEFINITION : In situation where several different outcomes are possible, we define the probability for any particular outcome as a fraction or proportion. If the possible outcomes are identified as A,B,C,D, and so on, then In situation where several different outcomes are possible, we define the probability for any particular outcome as a fraction or proportion. If the possible outcomes are identified as A,B,C,D, and so on, then Probability of A = number of outcomes classified as A Probability of A = number of outcomes classified as A total number of possible outcomes total number of possible outcomes p = fNp = fN

4. RANDOM SAMPLING DEFINITION : DEFINITION : A random sample must satisfy two requirements: A random sample must satisfy two requirements: 1- Each individual in the population has an equal chance of being selected. 1- Each individual in the population has an equal chance of being selected. 2- if more than one individual is to be selected for the sample, there must be constant probability for each and every selection. 2- if more than one individual is to be selected for the sample, there must be constant probability for each and every selection.

5. PROBABILITY AND THE NORMAL DISRIBUTION FIGURE 6.3 FIGURE 6.3 The normal distribution. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency). The equation is The normal distribution. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency). The equation is Y = 1 e -( x- μ ) 2 /2σ 2 Y = 1 e -( x- μ ) 2 /2σ 2 2πσ 2 2πσ 2

6. FIGURE 6.3

7. THE UNIT NORMAL TABLE FIGUER 6.5 FIGUER 6.5 The distribution for example 6.2 The distribution for example 6.2

8. PROBABILITY AND THE NORMAL DISRIBUTION Although the exact shape for the normal distribution is precisely defined by an equation (see figure 6.3), we can easily describe its general characteristics: Although the exact shape for the normal distribution is precisely defined by an equation (see figure 6.3), we can easily describe its general characteristics: 1- it is symmetrical. The left side is a mirror image of the right side, and the mean, median, and mode are equal. 1- it is symmetrical. The left side is a mirror image of the right side, and the mean, median, and mode are equal. 2- Fifty percent of the scores are below the mean, and 50 percent above it (mean = median) 2- Fifty percent of the scores are below the mean, and 50 percent above it (mean = median) 3-Most of the scores pile up around the mean (mean = mode), and extreme scores ( high or low) are relatively rare (low frequencies). 3-Most of the scores pile up around the mean (mean = mode), and extreme scores ( high or low) are relatively rare (low frequencies).

9. FIGURE 6.9 The distribution for Examples 6.4A-6.4B The distribution for Examples 6.4A-6.4B

10. ANSWERING PROBABILITY QUESTIONS WITH THE UNIT NORMAL TABLE FIGURE 6.10 FIGURE 6.10 The distribution for Examples 6.5. The distribution for Examples 6.5.

11. FIGURE 6.13 The distribution of SAT scores. The problem is to locate the score that separates the top 15% from the rest of the distribution. A line is drawn to divide the distribution roughly into 15% and 85% sections. The distribution of SAT scores. The problem is to locate the score that separates the top 15% from the rest of the distribution. A line is drawn to divide the distribution roughly into 15% and 85% sections.

12. PERCENTILES AND PERCENTILE RANKS FINDING PERCENTILE RANKS FINDING PERCENTILE RANKS FINDING PERCENTILES. FINDING PERCENTILES. FIGURE 6.16 FIGURE 6.16 The distribution for Example 6.9 The distribution for Example 6.9

13. QUARTILES Figure 6.17 Figure 6.17 The z-scores corresponding to the first, second, and third quartiles in a normal distribution The z-scores corresponding to the first, second, and third quartiles in a normal distribution

14. PROBABLITY AND THE BINOMINAL DISTRIBUTION THE BINOMINAL DISTRIBUTION THE BINOMINAL DISTRIBUTION 1-The two categories are identified as A and B. 1-The two categories are identified as A and B. 2- The probabilities (or proportions) associated with each category are identified as 2- The probabilities (or proportions) associated with each category are identified as P = p(A) = the probability of A P = p(A) = the probability of A q = p(B) = the probability of B q = p(B) = the probability of B Notice that p + q = 1.00 because A and B are the only two possible outcomes. Notice that p + q = 1.00 because A and B are the only two possible outcomes. 3- The number of individuals or observation in the sample is identified by n. 3- The number of individuals or observation in the sample is identified by n. 4-The variable X refers to the number of times category A occurs in the sample. 4-The variable X refers to the number of times category A occurs in the sample.

15. PROBABLITY AND THE BINOMINAL DISTRIBUTION DIFINITION DIFINITION Using the notation presented here, the binomial distribution shows the probability associated with each value of X from X = 0 to X = n. Using the notation presented here, the binomial distribution shows the probability associated with each value of X from X = 0 to X = n.

16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION Mean : μ = pn Mean : μ = pn Standard Deviation : σ = npq Standard Deviation : σ = npq Within this normal distribution, each value of X has a corresponding z-score, Within this normal distribution, each value of X has a corresponding z-score, z = x – μ = x-pn z = x – μ = x-pn σ npq σ npq

17. FIGURE 6.20 The relationship between the binomial distribution and the normal distribution. The relationship between the binomial distribution and the normal distribution.