1Chapter 6Continuous Probability Distributionsxf (x)Normalµσ2Continuous Probability Distributionsn Describes the possible outcomes and probabilities of occurrence for a continuous random variablen can't list all the possible values since there are an infinite number of values along the relevant portion of measurement scalen Are defined by an equation called a probability density function (PDF)n f(X)= [some equation involving X]3n The probability of the random variable assuming a value within the interval from x1to x2is equal to the relative area under the probability density function between x1and x2Continuous Probability Distributions: Conceptsn Probability = % of area under PDFn Total area beneath PDF = 1.0000n Area beneath a single point = 0.0000n Actually use calculus to find probabilitiesn We’ll use tables in bookxf (x)Normalx1x2µσ24NormalDistributionn A continuous probability distributionn Propertiesn unimodal & symmetrical (i.e., bell-shaped)n mean = median = moden ≈68% of observations within µ ± 1σn ≈95% of observations within µ ± 2σn ≈99% of observations within µ ± 3σxµ –3σ µ –1σµ –2σµ + 1 σµ + 2 σµ + 3 σµ68%95%99%n Prominent becausen many random variables roughly follow this pattern of distributionn it can be used to approximate Binomial, Poisson and other distributions under certain conditionsxf (x)Normalµσ5NormalDistributionn PDFn the height of the Normal curve at any point X once you supply the values for µ and σxf (x)Normalµσn where:n µ = meann σ = standard deviationn π = 3.14159n e = 2.718286NormalDistributionn The shape and location of any Normal distribution are determined by two parameters: µ and σn There is a different Normal curve for each set of values for µ and σ37Standard Normal Distributionn Allows the use of table for determining areas beneath any normal distribution, regardless of µ and σn We’ll denote as PN(x1 < x < x2| µ, σ)n Calculate the z-score for the endpoint(s) of the interval of interest, then use the z-score to look up the associated area(s) in the tablexf (x)Normalx1x2µσ8Calculating Normal Probabilitiesxf (x)Normalx1x2µσn You have an interval and need to determine the probability (area beneath the curve) contained within that intervaln determine µ & σ (givens)n sketch a normal curve and label µ & σn shade the interval of interest on the sketchn calculate the z-score for the interval’s endpointsn look up the area(s) for the z scoren determine the area beneath the curve for the intervaln Household Income9Calculating an X-value From a Known Probabilityn You have a probability (area beneath the curve) and need to determine the endpoint(s) that define the interval1. determine µ & σ (givens)2. sketch a Normal curve and label µ & σ3. determine the given probability (area beneath the curve) for the interval of interest4. shade the appropriate interval on the sketch5. use the area to look up the appropriate z-score6. use the z score, µ, & σ to back-solve for xn Household