Section 6'1 Introduction to the Normal Distribution

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Section 6.1Introduction to the Normal
Distribution
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Warm Ups Use Chebyshevs Theorem to complete
the following table
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Intro to the Normal Distribution

• The most important probability distribution in
  all of statistics.
• It is a continuous distribution (as opposed to
  discrete)
• Mainly studied by 2 mathematicians de Moivre and
  Gauss
• aka normal curve, bell-shaped curve, Gaussian
  distribution

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Properties of the Normal Distribution

• Highest point lies above the mean
• Bell-shaped
• Symmetric about the mean. This also implies that
  the mean and median are the same value.
• Tails approach the horizontal axis but never
  touch it (like an asymptote)
• Transition between concave down and concave up
  occurs at µ s and µ s

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Small s

• If s is small, the curve will be compact with a
  high peak

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Large s

• If s is large, the curve will be spread out with
  a low peak

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More Properties of the Normal

• The total area under any normal curve is 1. This
  is because the probabilities of any sample space
  sum to 1.
• The area under the curve for a given interval
  represents the probability that a measurement
  will lie in that interval.

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Chebyshev vs. Empirical

• Empirical Rule
• New
• This rule is good only for normal distributions.
  This rule is more specific and precise.
• 68 of the data falls in µ s
• 95 of the data falls in µ 2s
• 99.7 (nearly all) of the data falls in µ 3s

• Chebyshevs Theorem
• Review
• This theorem is good for any distribution no
  matter what the shape of the distribution is.
• At least 75 of the data falls in µ 2s
• At least 88.9 of the data falls in µ 3s
• At least 93.8 of the data falls in µ 4s

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Example

• The yearly wheat yield per acre on a particular
  farm is normally distributed with mean 35 bushels
  and standard deviation 8 bushels.
• What percentage of wheat yields are more than 35
  bushels?
• What percentage of wheat yields are between 19
  bushels and 51 bushels?
• What percentage of wheat yields are between 27
  bushels and 43 bushels?

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Example

• IQ Scores are approximately normally distributed
  with mean 100 and standard deviation 15.
• In what range of IQs does the middle 68 of the
  population lie?
• In what range of IQs does the middle 95 of the
  population lie?
• When Mrs. Brock was in school, in order to be
  classified as gifted a student had to have an IQ
  score of 130 or above. How may standard
  deviations above the mean is that? What percent
  of the population is gifted by these standards?