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Lesson 7 - 1

Lesson 7 - 1. Properties of the Normal Distribution. Quiz.

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Lesson 7 - 1

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  1. Lesson 7 - 1 Properties of the Normal Distribution

  2. Quiz • Homework Problem: Chapter 6 reviewThe number of cars that arrive at a bank’s window between 3:00 pm and 6 pm on Friday follows a Poisson process at the rate of 0.41 car every minute. Compute the possibility that the number of cars that arrive at the bank between 4:00 pm and 4:10 pm isa) exactly four carsb) fewer than four carsc) at least four cars • Reading questions: • What does the area under the graph represent in a continuous PDF? • What is the standard normal distribution variable called?

  3. Objectives • Understand the uniform probability distribution • Graph a normal curve • State the properties of the normal curve • Understand the role of area in the normal density function • Understand the relationship between a normal random variable and a standard normal random variable

  4. 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!!)

  5. Uniform PDF • Sometimes we want to model a random variable that is equally likely between two limits • When “every number” is equally likely in an interval, this is a uniformprobabilitydistribution • Any specific number has a zero probability of occurring • The mathematically correct way to phrase this is that any two intervals of equal length have the same probability • Examples • Choose a random time … the number of seconds past the minute is random number in the interval from 0 to 60 • Observe a tire rolling at a high rate of speed … choose a random time … the angle of the tire valve to the vertical is a random number in the interval from 0 to 360

  6. Discrete Uniform PDF P(x=0) = 0.25 P(x=1) = 0.25 P(x=2) = 0.25 P(x=3) = 0.25 Continuous Uniform PDF P(x=1) = 0 P(x ≤ 1) = 0.33 P(x ≤ 2) = 0.66 P(x ≤ 3) = 1.00

  7. Probability in a Continuous Probability Distributions Let P(x) denote the probability that the random variable X equals x, then 1) ∑ P(x) = 1 (sum of all probabilities must equal 1) → total area under the PDF graph must equal 1 2) The probability of x occurring in any interval, P(x), must between 0 and 1 0 ≤ P(x) ≤ 1 → the height of the graph of the PDF must be greater than or equal to 0 for all possible values of the random variable 3) The area underneath probability density function over some interval represents the probability of observing a value of the random variable in that interval.

  8. 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

  9. 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

  10. 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 Normal Probability Density Function 1 y = -------- e √2π -(x – μ)2 2σ2 where μ is the mean and σ is the standard deviation of the random variable x

  11. 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]

  12. 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 Note: we are going to use tables (for Z statistics) not the normal PDF!! Or our calculator (see next chart)

  13. Normal Distributions on TI-83 • normalpdfpdf = Probability Density FunctionThis function returns the probability of a single value of the random variable x.  Use this to graph a normal curve.Using this function returns the y-coordinates of the normal curve. • Syntax:  normalpdf (x, mean, standard deviation)taken from http://mathbits.com/MathBits/TISection/Statistics2/normaldistribution.htm

  14. 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 the lower bound. • Syntax:  normalcdf (lower bound, upper bound, mean, standard deviation)(note: lower bound is optional and we can use -E99 for negative infinity and E99 for positive infinity)

  15. 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)

  16. Example 1 A random number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform distribution • Draw a graph of this distribution • What is the P(0<X<0.2)? • What is the P(0.25<X<0.6)? • What is the probability of getting a number > 0.95? • Use calculator to generate 200 random numbers 1 0.20 1 0.35 0.05 Math  prb  rand(200) STO L3 then 1varStat L3

  17. Example 2 A random variable x is normally distributed with μ=10 and σ=3. • Compute Z for x1 = 8 and x2 = 12 • If the area under the curve between x1 and x2 is 0.495, what is the area between z1 and z2? 8 – 10 -2 Z = ---------- = ----- = -0.67 3 3 12 – 10 2 Z = ----------- = ----- = 0.67 3 3 0.495

  18. Summary and Homework • Summary • Normal probability distributions can be used to model data that have bell shaped distributions • Normal probability distributions are specified by their means and standard deviations • Areas under the curve of general normal probability distributions can be related to areas under the curve of the standard normal probability distribution • Homework • pg 367 – 371; 7 – 12; 15-16, 19-20, 32-33

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