A32 APPENDIX D Key Formulas CHAPTER 3 Classical (or Theoretical) Probability: P(E) = Number of outcomes in event E Total number of outcomes in sample space Empirical (or Statistical) Probability: P(E) = Frequency of event E Total frequency = f n Probability of a Complement: P(E′) = 1 - P(E) Probability of occurrence of both events A and B: P(A and B) = P(A) # P(B | A) P(A and B) = P(A) # P(B) if A and B are independent Probability of occurrence of either A or B: P(A or B) = P(A) + P(B) - P(A and B) P(A or B) = P(A) + P(B) if A and B are mutually exclusive Permutations of n objects taken r at a time: nPr = n! (n - r)! , where r … n Distinguishable Permutations: n1 alike, n2 alike, . . . , nk alike: n! n1! # n2! # n3! . . . nk! , where n1 + n2 + n3 + . . . + n k = n Combinations of n objects taken r at a time: nCr = n! (n - r)!r! , where r … n CHAPTER 4 Mean of a Discrete Random Variable: m = gxP(x) Variance of a Discrete Random Variable: s 2 = g(x - m) 2P(x) Standard Deviation of a Discrete Random Variable: s = 2s 2 = 2g(x - m) 2P(x) Expected Value: E(x) = m = gxP(x) Binomial Probability of x successes in n trials: P(x) = nCxp xqn-x = n! (n - x)!x! pxqn-x Population Parameters of a Binomial Distribution: Mean: m = np Variance: s 2 = npq Standard Deviation: s = 2npq Geometric Distribution: The probability that the first success will occur on trial number x is P(x) = pqx-1, where q = 1 - p. Poisson Distribution: The probability of exactly x occurrences in an interval is P(x) = m xe-m x! , where e ≈ 2.71828 and m is the mean number of occurrences per interval unit. CHAPTER 5 Standard Score, or z-Score: z = Value - Mean Standard deviation = x - m s Transforming a z-Score to an x-Value: x = m + zs Central Limit Theorem (n Ú 30 or population is normally distributed): Mean of the Sampling Distribution: mx = m Variance of the Sampling Distribution: s 2 x = s 2 n Standard Deviation of the Sampling Distribution (Standard Error): sx = s2 n z-Score = Value - Mean Standard Error = x - mx sx = x - m s/2n
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