Business Statistics: A Decision-Making Approach 7th Edition Chapter Introduction to Continuous Probability Distributions Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-1 Chapter Goals After completing this chapter, you should be able to: Convert values from any normal distribution to a standardized z-score Find probabilities using a normal distribution table Apply the normal distribution to business problems Recognize when to apply the uniform and exponential distributions Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-2 Probability Distributions Probability Distributions Ch Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson Uniform Business Statistics: A DecisionHypergeometric Making Approach, 7e © 2008 Prentice-Hall, Inc Ch Exponential Chap 6-3 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure Business Statistics: A Decisionaccurately Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-4 The Normal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Exponential Chap 6-5 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an Business Statistics: A Decisioninfinite theoretical range: Making Approach, 7e © 2008 + to Prentice-Hall, Inc f(x) σ μ Mean = Median = Mode Chap 6-6 x Many Normal Distributions By varying the parameters μ and σ, we obtain Business Statistics: A Decisiondifferent normal distributions Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-7 The Normal Distribution Shape f(x) Changing μ shifts the distribution left or right σ Business Statistics: A Decision- μ Making Approach, 7e © 2008 Prentice-Hall, Inc Changing σ increases or decreases the spread x Chap 6-8 Finding Normal Probabilities Probability is measured by the area under the curve f(x) Business Statistics: A Decision- a Making Approach, 7e © 2008 Prentice-Hall, Inc P (a x b) b x Chap 6-9 Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(x) P( x μ) 0.5 0.5 Business Statistics: A Decision- μ Making Approach, 7e © P(2008 Prentice-Hall, Inc P(μ x ) 0.5 0.5 x x ) 1.0 Chap 6-10 PHStat Dialogue Box Business Statistics:Select A Decisiondesired options Making Approach, 7e and © 2008 enter values Prentice-Hall, Inc Chap 6-32 PHStat Output Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-33 The Uniform Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Exponential Chap 6-34 The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-35 The Uniform Distribution (continued) The Continuous Uniform Distribution: f(x) = b a if a x b otherwise where f(x) = value of the density function at any x value a = lower limit of the interval Statistics: A Decisionb = upper limit of the interval Business Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-36 The Mean and Standard Deviation for the Uniform Distribution The mean (expected value) is: a b E(x) μ The standard deviation is (b a)2 σ 12 where a = lower limit of the interval from a to b Business Statistics: A Decisionb = upper of the interval from a to b Making Approach, 7e ©limit 2008 Prentice-Hall, Inc Chap 6-37 Uniform Distribution Example: Uniform Probability Distribution Over the range ≤ x ≤ 6: f(x) = - = 25 for ≤ x ≤ f(x) 25 Business Statistics: A Decision2 Making Approach, 7e © 2008 Prentice-Hall, Inc x Chap 6-38 Uniform Distribution Example: Uniform Probability Distribution Over the range ≤ x ≤ 6: 26 E(x) μ 4 2 (b a) (6 2) σ 1.1547 12 12 Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-39 The Exponential Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Exponential Chap 6-40 The Exponential Distribution Used to measure the time that elapses between two occurrences of an event (the time between arrivals) Examples: Time between trucks arriving at an unloading dock Time between transactions at an ATM Machine Time Business Statistics: A between Decision- phone calls to the main operator Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-41 The Exponential Distribution The probability that an arrival time is equal to or less than some specified time a is P(0 x a) 1 e λa where 1/ is the mean time between events Note that if the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential Business Statistics: A Decisionwith mean time 1/ Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-42 Exponential Distribution (continued) Shape of the exponential distribution f(x) = 3.0 (mean = 333) = 1.0 (mean = 1.0) = 0.5 (mean = 2.0) Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc x Chap 6-43 Example Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed) What is the probability that the arrival time between consecutive customers is less than five minutes? Time between arrivals is exponentially distributed with mean time between arrivals of minutes (15 per 60 minutes, on average) 1/ = 4.0, so = 25 -a -(.25)(5) P(x < 5) = e = – e = 7135 Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-44 Using PHStat Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-45 Chapter Summary Reviewed key continuous distributions normal uniform exponential Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 6-46 ... Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an Business Statistics: A Decisioninfinite theoretical... Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation: x μ z σ Business Statistics: A DecisionMaking Approach, 7e © 2008... Prentice-Hall, Inc f(x) σ μ Mean = Median = Mode Chap 6-6 x Many Normal Distributions By varying the parameters μ and σ, we obtain Business Statistics: A Decisiondifferent normal distributions Making Approach,