Statistics 270 - Lecture 14

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Statistics 270 - Lecture 14
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Example

• Consider a rv, X, with pdf
• Sketch pdf

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Example

• cdf of X
• Find E(X)

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Normal Distributions

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

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Normal Distributions

• Density
• cdf

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

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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)

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Example

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

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Example

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

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Standardizing

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

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• 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?

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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

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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?