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Lesson 7 - QR

Lesson 7 - QR. Quiz Review. Objectives. Review for the chapter 7 quiz on sections 7-1 through 7-3. Vocabulary. Continuous random variable – has infinitely many values

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Lesson 7 - QR

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  1. Lesson 7 - QR Quiz Review

  2. Objectives • Review 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 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!!)

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

  5. 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

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

  7. 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

  8. 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]

  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 FunctionThis 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 • invNorminv = Inverse Normal PDFThis 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 a a a b

  13. 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

  14. Problems • Standard Normal Random Variable What is the Z value associate with 91st 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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