CSC 323 Quarter: Spring

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CSC 323 Quarter Spring 02/03

• Daniela Stan Raicu
• School of CTI, DePaul University

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Outline

• Normal Distributions
• The Empirical (68-95-99.7) rule
• Standard Normal Distribution
• Normal Distributions Calculations
• Introduction to the Statistical Software SAS

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The normal distributions

• Normal curves are density curves that are
• Symmetric
• Unimodal
• Bell-Shaped

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The normal distributions (cont.)

• A normal distribution is specified by
• Mean ?
• Standard Deviation ?
• Notation N(?, ?)

• The equation of the normal
• distribution ( gives the height of
• the normal distribution)

f(x)
Example of two normal curves specified by their
mean and standard deviation
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The Empirical Rule for Any Normal Curve
68
95
?
?1 ?
?
?2 ?
?-2 ?
?-1 ?
99.7
?
?3 ?
?-3 ?
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The 68-95-99.7 (empirical) rule

• In the normal distribution N(?, ?)
• Approximately 68 of the observations are between
  ?- ? and ? ?
• Approximately 95 of the observations are between
  ?- 2? and ? 2?
• Approximately 99.7 of the observations are
  between ?- 3? and ? 3?

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Example

• The heights of adult women in the United States
  follow, at least approximately, a bell-shaped
  curve. What do you think that means?

The most adult women are clumped around the
average, with numbers decreasing the farther
values are from the average in either direction.
Health and Nutrition Examination Study of
1976-1980 (HANES)

• The average of the heights of adult women is ?
  65 and the standard deviation is ? 2.5. What
  does the 68-95-97.7 rule imply?

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The empirical rule
65
652.5
65-2.5

• 68 of adult women have heights between 62.5 and
  67.5 inches
• 95 of adult women have heights between 60 and
  70 inches
• 99.7 of adult women have heights between 67.5
  and 72.5 inches

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Health and Nutrition Examination Study of
1976-1980 (HANES)

• What proportion of individuals fall into any
  range of values?
• Example What proportion of men are less than
  68 inches tall?
• At what percentile a given individual falls, if
  you know their values
• What value corresponds to a given percentile

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Standardized score

• A standardized score is simply the number of
  standard deviations an individual falls above or
  below the mean for the whole group.
• Values above the mean have positive standardized
  scores values below the mean have negative ones.
• Example
• Females (ages 18-24) have a mean height of 65
  inches and a standard deviation of 2.5 inches.
  What is the standardized score of a a women who
  is 67.5 inches tall?
• Standardized score
• (67.5 65)/2.51

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Standardized Scores

• standardized score
• (observed value - mean) / (std dev)
• z is the standardized score
• x is the observed value
• m is the population mean
• s is the population standard deviation

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The standard normal distribution

• The standard normal distribution N(0,1) is the
  normal distribution with mean 0 and standard
  deviation 1

• If a variable X has any normal distribution N(?,
  ?), then the standardized variable Z

has the standard normal distribution N(0,1).
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Normal distribution calculations
Example The heights of young women are
approximately normal with mean ?64.5 inches and
?2.5 inches. What is the proportion of women
how are less than 68 inches tall?
1. State the problem X height, X lt 68
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Normal distribution calculations
3. What proportion of observations/women on the
standard normal variable Z take values less than
1.4?
Table entry is area to the left of z
Area (Zlt1.4).9192
Table A at the end of the book gives areas
(proportions of observations) under standard
normal curve.