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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-1 Introduction to Statistics Chapter 5 Random Variables

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-2 Chapter Goals After completing this chapter, you should be able to: Understand the discrete random variables and there probability distribution. Compute the expected value and standard deviation for a discrete probability distribution Apply the binomial distribution to applied problems

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-3 Introduction to Probability Distributions Random Variable Represents a possible numerical value from a random event Random Variables Discrete Random Variable Continuous Random Variable

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-4 A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes: number of complaints per day number of TV’s in a household number of rings before the phone is answered Only two possible outcomes: gender: male or female defective: yes or no spreads peanut butter first vs. spreads jelly first Discrete Probability Distributions

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-5 Discrete Random Variables Can only assume a countable number of values Examples: Roll a die twice Let x be the number of times 4 comes up (then x could be 0, 1, or 2 times) Toss a coin 5 times. Let x be the number of heads (then x = 0, 1, 2, 3, 4, or 5)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-6 Experiment: Toss 2 Coins. Let x = # heads. T T Discrete Probability Distribution 4 possible outcomes T T H H HH Probability Distribution 0 1 2 x x Value Probability 0 1/4 =.25 1 2/4 =.50 2 1/4 =.25.50.25 Probability

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-7 A list of all possible [ x i, P(x i ) ] pairs x i = Value of Random Variable (Outcome) P(x i ) = Probability Associated with Value x i ’s are mutually exclusive (no overlap) x i ’s are collectively exhaustive (nothing left out) 0 P(x i ) 1 for each x i P(x i ) = 1 Discrete Probability Distribution

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-8 Discrete Random Variable Summary Measures Expected Value of a discrete distribution (Weighted Average) E(x) = x i P(x i ) Example: Toss 2 coins, x = # of heads, compute expected value of x: E(x) = (0 x.25) + (1 x.50) + (2 x.25) = 1.0 x P(x) 0.25 1.50 2.25

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-9 Standard Deviation of a discrete distribution where: E(x) = Expected value of the random variable x = Values of the random variable P(x) = Probability of the random variable having the value of x Discrete Random Variable Summary Measures (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-10 Example: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1) Discrete Random Variable Summary Measures (continued) Possible number of heads = 0, 1, or 2

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-11 Two Discrete Random Variables Expected value of the sum of two discrete random variables: E(x + y) = E(x) + E(y) = x P(x) + y P(y) (The expected value of the sum of two random variables is the sum of the two expected values)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-12 The Binomial Distribution Characteristics of the Binomial Distribution: A trial has only two possible outcomes – “success” or “failure” There is a fixed number, n, of identical trials The trials of the experiment are independent of each other The probability of a success, p, remains constant from trial to trial If p represents the probability of a success, then (1-p) = q is the probability of a failure

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-13 Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for a contract will either get the contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-14 Counting Rule for Combinations A combination is an outcome of an experiment where x objects are selected from a group of n objects where: n! =n(n - 1)(n - 2)... (2)(1) x! = x(x - 1)(x - 2)... (2)(1) 0! = 1 (by definition)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-15 P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of ‘successes’ in sample, (x = 0, 1, 2,..., n) p = probability of “success” per trial q = probability of “failure” = (1 – p) n = number of trials (sample size) P(x) n x ! nx pq x n x ! ()! Example: Flip a coin four times, let x = # heads: n = 4 p = 0.5 q = (1 -.5) =.5 x = 0, 1, 2, 3, 4 Binomial Distribution Formula

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-16 n = 5 p = 0.1 n = 5 p = 0.5 Mean 0.2.4.6 012345 X P(X).2.4.6 012345 X P(X) 0 Binomial Distribution The shape of the binomial distribution depends on the values of p and n Here, n = 5 and p =.1 Here, n = 5 and p =.5

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-17 Binomial Distribution Characteristics Mean Variance and Standard Deviation Wheren = sample size p = probability of success q = (1 – p) = probability of failure

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-18 n = 5 p = 0.1 n = 5 p = 0.5 Mean 0.2.4.6 012345 X P(X).2.4.6 012345 X P(X) 0 Binomial Characteristics Examples

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-19 Using Binomial Tables n = 10 xp=.15p=.20p=.25p=.30p=.35p=.40p=.45p=.50 0 1 2 3 4 5 6 7 8 9 10 0.1969 0.3474 0.2759 0.1298 0.0401 0.0085 0.0012 0.0001 0.0000 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010 10 9 8 7 6 5 4 3 2 1 0 p=.85p=.80p=.75p=.70p=.65p=.60p=.55p=.50x Examples: n = 10, p =.35, x = 3: P(x = 3|n =10, p =.35) =.2522 n = 10, p =.75, x = 2: P(x = 2|n =10, p =.75) =.0004

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-20 Chapter Summary Distinguished between discrete and continuous probability distributions Examined discrete probability distributions and their summary measures Apply the binomial distribution to applied problems

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