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Statistics 270 - Lecture 14

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Statistics 270 - Lecture 14. Example. Consider a rv, X, with pdf. Sketch pdf. Example ... Common continuous density is the normal distribution. It is symmetric, ... – PowerPoint PPT presentation

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Title: Statistics 270 - Lecture 14


1
Statistics 270 - Lecture 14
2
Example
  • Consider a rv, X, with pdf
  • Sketch pdf

3
Example
  • cdf of X
  • Find E(X)

4
Normal Distributions
  • Common continuous density is the normal
    distribution
  • It is symmetric, bell-shaped and uni-modal
  • Denoted

5
Normal Distributions
  • Density
  • cdf

6
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7
  • What happens if mean is changed?
  • What happens if standard deviation is changed?

8
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9
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10
Standard Normal
  • The standard normal distribution is a particular
    normal distributrion
  • XN(0,1)
  • pdf
  • Have table of cumulative probabilities for
    standard normal (Table A-3)

11
Example
  • Suppose Z has a standard nomral distribution.
    Find
  • P(Zlt1.96)
  • P(Zlt3.02)
  • P(Zgt3.03)

12
Example
  • Suppose Z has a standard nomral distribution.
    Find
  • P(Zlt3.025)

13
Standardizing
  • If X is any R.V., the standardized variable, Z,
    has mean 0 and standard deviation 1

14
  • Distribution of scores on a standardized test can
    be approximated by a normal distribution with
    mean of 500 and standard deviation of 100. Find
    probability that a randomly selected student
    scores
  • Over 650
  • Between 325 and 675
  • What proportion of students score better than 680?

15
68-95-99.7 Rule
  • For a random variable, X, that is normally
    distributed with mean, m, and standard deviation,
    s
  • 68 of the observations will fall within 1
    standard deviation of the mean
  • 95 of the observations will fall within 2
    standard deviation of the mean
  • 99.7 of the observations will fall within 3
    standard deviation of the mean

16
Example (page 65)
  • The distribution of heights of young women aged
    20-29 is approximately Normally distributed with
    mean 64 inches and standard deviation 2.7 inches
  • Between what heights do 95 of the heights of
    young women fall?
  • What percent of young women are taller than 61.3
    inches?
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