Math 3680

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Math 3680 Lecture 10 Normal Random Variables
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• Many distributions follow the bell curve rather
  well. For example, according to studies, the
  height of U.S. women has the following data m
  63.5 inches and s 2.5 inches
• Accordingly, about 68 of women have heights
• between 61 and 66 inches.

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• However, the bell curve does not model all data
  sets well.

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• Example What percentage of women has heights
  between 60 and 68 inches?

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• Example What percentage of women has heights
  between 60 and 68 inches?
• Is this answer exact or an approximation?

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• Example For a certain population of high school
  students, the SAT-M scores are normally
  distributed with m 500 and s 100. A
  certain engineering college will accept only high
  school seniors with SAT-M scores in the top 5.
  What is the minimum SAT-M score for this program?
  
• Note Before, the kind of question that was
  posed was What percentage of students score
  above 700 on the SAT-M? Now, the percentage is
  specified.

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• Approximating a Binomial(n, p) distribution with
  a normal curve

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• Example From a heterozygous cross of two pea
  plants, 192 seeds are planted. According to the
  laws of genetics, there is a 25 chance that any
  one offspring will be recessive homozygous,
  independently of all other offspring.
• Find the probability that between 44 and 55
  (inclusive) of the seeds are recessive homozygous.

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• Solution 1 (Exact). Let X denote the number
  of recessive homozygous offspring. Then
• X Binomial(192, 0.25)
• Therefore,

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In terms of the probability histogram
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• Solution 2 (Approximate). Note that the
  probability histogram of the Binomial(192, 0.25)
  distribution is approximated well by a bell
  curve. Since
• X Binomial(192, 0.25),
• we have
• We convert 43.5 and 55.5 to standard units

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• Solution 2 (continued)
• Therefore,
• Question Why did we standardize 43.5 and 55.5?
• This subtlety is called continuity correction.

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BBinomial(n, 0.5) distribution
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BBinomial(n, 0.5) distribution
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BBinomial(100, p) distribution
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BBinomial(100, p) distribution
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BBinomial(n,p) distribution with p small
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• Limitations On Normal Approximation
• The normal approximation is reasonable if n is
  large and both p and 1 - p are not small.
  More precisely, the approximation is good if
• n p ? 5 and n (1 - p) ? 5.
• What happens if n is large and p is small?
• What if instead p is close to 1?

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• Continuity Correction
• The primary difficulty with using the continuity
  correction is deciding whether to add or subtract
  0.5 from the endpoints. This decision may be
  facilitated by drawing a rough histogram and then
  deciding which rectangles are to be included.
• Example In the previous problem, what should be
  converted to standard units to find
• P(24 ? X ? 30)
• P(31 lt X ? 38)
• P(X gt 25)
• P(X ? 37)

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• Continuity Correction
• Theres nothing particularly special about the
  binomial distribution for this procedure.
• The continuity correction can be used whenever a
  discrete random variable is being approximated by
  a continuous one.

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• Example A gambler repeatedly bets on red in
  roulette. The chance of winning on one play is p
  9/19. Suppose the gambler plays 100 times. Use
  the normal approximation to estimate the
  probability that the gambler wins more than 50
  times.