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Lesson 7 - QR Quiz Review

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Objectives Review for the chapter 7 quiz on sections 7-1 through 7-3

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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 X Normal curve – bell shaped curve Normal distributed random variable – has a PDF or relative frequency histogram shaped like a normal curve Standard normal – normal PDF with mean of 0 and standard deviation of 1 (a z statistic!!)

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P(x=1) = 0 P(x ≤ 1) = 0.33 P(x ≤ 2) = 0.66 P(x ≤ 3) = 1.00 Continuous Uniform PDF Since 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.

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Properties of the Normal Density Curve It is symmetric about its mean, μ Because mean = median = mode, the highest point occurs at x = μ It has inflection points at μ – σ and μ + σ Area under the curve = 1 Area under the curve to the right of μ equals the area under the curve to the left of μ, which equals ½ As 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

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Empirical Rule μ μ - σ μ - 2σ μ - 3σμ + σ μ + 2σ μ + 3σ 34% 13.5% 2.35% 0.15% μ ± σ μ ± 2σ μ ± 3σ 68% 95% 99.7%

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Normal Curves Two normal curves with different means (but the same standard deviation) [on left] –The curves are shifted left and right Two normal curves with different standard deviations (but the same mean) [on right] –The curves are shifted up and down

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Area under a Normal Curve The area under the normal curve for any interval of values of the random variable X represents either The proportion of the population with the characteristic described by the interval of values or The 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]

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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

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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)

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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)

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ApproachGraphicallySolution Find the area to the left of z a P(Z < a) Shade the area to the left of z a Use Table IV to find the row and column that correspond to z a. The area is the value where the row and column intersect. Normcdf(-E99,a,0,1) Find the area to the right of z a P(Z > a) or 1 – P(Z < a) Shade the area to the right of z a Use Table IV to find the area to the left of z a. The area to the right of z a is 1 – area to the left of z a. Normcdf(a,E99,0,1) or 1 – Normcdf(-E99,a,0,1) Find the area between z a and z b P(a < Z < b) Shade the area between z a and z b Use Table IV to find the area to the left of z a and to the left of z a. The area between is area zb – area za. Normcdf(a,b,0,1) Obtaining Area under Standard Normal Curve aa a b

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Problems Standard Normal Random Variable P(Z < 1.96) = 0.975 normalcdf(-E99,1.96) P(Z > 0.57) = 0.284 normalcdf(0.57,E99) P(-2.71 < Z < 1.09) = 0.859 normalcdf(-2.71,1.09) Regular Normal Random Variable P(x<4) = 0.965 normalcdf(-E99,4,2,1.1) with μ=2 σ=1.1 P(x>16) = 0.965 normalcdf(16,E99,10,3.84) with μ=10 σ=3.84

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Problems Standard Normal Random Variable What is the Z value associate with 91 st percentile? Z = invNorm(0.91) = Regular Normal Random Variable What is the X value associated with 57% to the right with μ = 11 and σ = 3? X = invNorm(1-0.57,11,3) = 10.47 invNorm uses area to the left!

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