2 ObjectivesReview for the chapter 7 quiz on sections 7-1 through 7-3
3 Vocabulary Continuous random variable – has infinitely many values Uniform probability distribution – probability distribution where the probability of occurrence is equally likely for any equal length intervals of the random variable XNormal curve – bell shaped curveNormal distributed random variable – has a PDF or relative frequency histogram shaped like a normal curveStandard normal – normal PDF with mean of 0 and standard deviation of 1 (a z statistic!!)
4 Continuous Uniform PDF P(x=1) = 0P(x ≤ 1) = 0.33P(x ≤ 2) = 0.66P(x ≤ 3) = 1.00Since the area under curve must equal one. The height or P(x) will always be equal to 1/(b-a), where b is the upper limit and a the lower limit. Probabilities are just the area of the appropriate rectangle.
5 Properties of the Normal Density Curve It is symmetric about its mean, μBecause mean = median = mode, the highest point occurs at x = μIt has inflection points at μ – σ and μ + σArea under the curve = 1Area under the curve to the right of μ equals the area under the curve to the left of μ, which equals ½As x increases or decreases without bound (gets farther away from μ), the graph approaches, but never reaches the horizontal axis (like approaching an asymptote)The Empirical Rule applies
7 Normal CurvesTwo normal curves with different means (but the same standard deviation) [on left]The curves are shifted left and rightTwo normal curves with different standard deviations (but the same mean) [on right]The curves are shifted up and down
8 Area under a Normal Curve The area under the normal curve for any interval of values of the random variable X represents eitherThe proportion of the population with the characteristic described by the interval of values orThe probability that a randomly selected individual from the population will have the characteristic described by the interval of values [the area under the curve is either a proportion or the probability]
9 Standardizing a Normal Random Variable our Z statistic from before X - μ Z = σ where μ is the mean and σ is the standard deviation of the random variable X Z is normally distributed with mean of 0 and standard deviation of 1 Z measures the number of standard deviations away from the mean a value of X is
10 Normal Distributions on TI-83 normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set a different lower bound.Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) (note: we use -E99 for negative infinity and E99 for positive infinity)
11 Normal Distributions on TI-83 invNorm inv = Inverse Normal PDF This function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation.Syntax: invNorm (probability, mean, standard deviation)
12 Obtaining Area under Standard Normal Curve ApproachGraphicallySolutionFind the area to the left of zaP(Z < a)Shade the area to the left of zaUse Table IV to find the row and column that correspond to za. The area is the value where the row and column intersect.Normcdf(-E99,a,0,1)Find the area to the right of zaP(Z > a) or1 – P(Z < a)Shade the area to the right of zaUse Table IV to find the area to the left of za. The area to the right of za is 1 – area to the left of za.Normcdf(a,E99,0,1) or1 – Normcdf(-E99,a,0,1)Find the area between za and zbP(a < Z < b)Shade the area between za and zbUse Table IV to find the area to the left of za and to the left of za. The area between is areazb – areaza.Normcdf(a,b,0,1)aaab
13 Problems Standard Normal Random Variable P(Z < 1.96) = normalcdf(-E99,1.96)P(Z > 0.57) = normalcdf(0.57,E99)P(-2.71 < Z < 1.09) = normalcdf(-2.71,1.09)Regular Normal Random VariableP(x<4) = normalcdf(-E99,4,2,1.1)with μ=2 σ=1.1P(x>16) = normalcdf(16,E99,10,3.84)with μ=10 σ=3.84
14 Problems Standard Normal Random Variable What is the Z value associate with 91st percentile?Z = invNorm(0.91) =Regular Normal Random VariableWhat is the X value associated with 57% to the right with μ = 11 and σ = 3?X = invNorm(1-0.57,11,3) = 10.47invNorm uses area to the left!