Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 16Normal Distributions

• Everything is Back to Normal (Almost)

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Normal DistributionsOutline/learning Objectives

• To identify and describe an approximately normal
  distribution.
• To state properties of a normal distribution.
• To understand a data set in terms of standardized
  data values.

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Normal DistributionsOutline/learning Objectives

• To state the 68-95-99.7 rule.
• To apply the honest and dishonest-coin principles
  to understand the concept of a confidence
  interval.

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Normal Distributions

• 16.1 Approximately Normal Distributions of Data

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Normal Distributions

• Approximately normal distribution
• Data sets that can be described as having bar
  graphs that roughly fit a bell-shaped pattern.
• Normal distribution
• A distribution of data that has a perfect bell
  shape.
• Normal curves
• Perfect bell-shaped curves.

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Normal Distributions

• 16.2 Normal Curves and Normal Distributions

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Normal Distributions

• Symmetry
• Every normal curve has a vertical axis of
  symmetry, splitting the bell-shaped region
  outlined by the curve into two identical halves.
  We can refer to it as the line of symmetry.

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Normal Distributions

• Median/mean.
• We call the point of intersection of the
  horizontal axis and the line of symmetry of the
  curve the center of the distribution. The center
  is both the median and the mean (average) of the
  data. We use the Greek letter ? (mu) to denote
  this value.

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Normal Distributions

• Median and Mean of a Normal Distribution
• In a normal distribution, M ? . (If the
  distribution is approximately normal, then M ?
  ?).
• The fact that the median equals the mean implies
  that 50 of the data are less than or equal to
  the mean and 50 of the data are greater than or
  equal to the mean. For data fitting an
  approximately normal distribution, the median and
  the mean should be close to each other but we
  should not expect them to be equal.

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Normal Distributions

• Standard deviation.
• The easiest way to describe the standard
  deviation of a normal distribution is to look at
  the normal curve. If you bend a piece of wire
  into a bell-shaped normal curve at the very top,
  you would be bending the wire downward (a),

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Normal Distributions

• Standard deviation.
• but at the bottom you would be bending the wire
  upward (b). As you move your hands down the
  wire, the curvature gradually changes, and there
  is one point on each side of

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Normal Distributions

• Standard deviation.
• the curve where the transition from being bent
  downward to being bent upward takes place. Such
  a point P (in figure c) is called a point of
  inflection of the curve.

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Normal Distributions

• The standard deviation of a normal distribution
  is the horizontal distance between the line of
  symmetry of the curve and one of the two points
  of inflection (P or P' )

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Normal Distributions

• Standard Deviation of a Normal Distribution
• In a normal distribution, the standard deviation
  ? equals the distance between a point of
  inflection and the line of symmetry of the curve.
• Quartiles of a Normal Distribution
• In a normal distribution, Q3 ? ? (0.675) ?
  and Q1 ? ? (0.675) ?.

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Normal Distributions

• 16.3 Standardizing Normal Data

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Normal Distributions

• Standardizing
• To standardize a data value x, we measure how
  far x has strayed from the mean ? using the
  standard deviation ? as the unit of measurement.
• Z-value
• A standardized data value.

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Normal Distributions

• Standardizing Rule
• In a normal distribution with mean ? and
  standard deviation ? , the standardized value of
  a data point x is z (x - ?)/? .

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Normal Distributions

• From x to z Part 2
• A normal distributed data set with mean ?
  63.18lb and standard deviation ? 13.27lb. What
  is the standardized value of x 91.54lb?
• z (91.54 63.18)/13.27 2.13715 ? 2.14

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Normal Distributions

• 16.4 The 68-95-99.7 Rule

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Normal Distributions

• The 68-95-99.7 Rule
• 1. In every normal distribution, about 68 of all
  the data values fall within one standard
  deviation above and below the mean. In other
  words, 68 of all the data have standardized
  values between z -1 and z 1.

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Normal Distributions

• The 68-95-99.7 Rule
• 1 (cont). The remaining 32 are divided equally
  between data with standardized values z ? -1 and
  data with standardized values z ? 1 (see figure
  a).

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Normal Distributions

• The 68-95-99.7 Rule
• 2. In every normal distribution, about 95 of all
  the data values fall within two standard
  deviations above and below the mean. In other
  words, 95 of all the data have standardized
  values between z -2 and z 2.

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Normal Distributions

• The 68-95-99.7 Rule
• 2 (cont). The remaining 5 of the data are
  divided equally between data with standardized
  values z ? -2 and data with standardized values z
  ? 2. (see figure b).

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Normal Distributions

• The 68-95-99.7 Rule
• 3. In every normal distribution, about 99.7
  (practically 100) of all the data values fall
  within three standard deviations above and below
  the mean. In other words, 99.7 of all the data
  have standardized values between z -3 and z
  3. There is a minuscule amount of data with
  standardized values outside the range (see figure
  b).

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Normal Distributions

• 16.5 Normal Curves as Models of Real-Life Data
  Sets

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Normal Distributions

• The 68-95-99.7 Rule for Normal curves
• About 68 of the data values fall within (plus or
  minus) one standard deviation of the mean.
• About 95 of the data values fall within (plus or
  minus) two standard deviations of the mean.
• About 99.7, or practically 100, of the data
  values fall within (plus or minus) three standard
  deviations of the mean.

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Normal Distributions

• 16.6 Distributions of Random Events

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Normal Distributions

• Coin-Tossing Experiments Part 1
• Distribution of random variable X (number of
  Heads in 100 coin tosses) (a) 10 times, (b) 100
  times, (c) 500 times, (d) 1000 times, (e) 5000
  times, and (f) 10,000 times.

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Normal Distributions

• 16.7 Statistical Inference

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Normal Distributions

• The Honest-Coin Principle
• Suppose an honest coin is tossed n times (n ?
  30), and let X denote the number of Heads that
  come up. The random variable X has an
  approximately normal distribution with mean ?
  n/2 Heads and standard deviation heads
  .

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Normal Distributions

• Coin-Tossing Experiments Part 2
• An honest coin is going to be tossed 256 times.
  Lets say that we can make a bet that if the
  number of Heads tossed falls somewhere between
  120 and 136, we will win otherwise we will lose.
  Should we make such a bet?

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Normal Distributions

• Coin-Tossing Experiments Part 2
• X 256
• ? n/2 256/2 128
• The values 120 to 136 are exactly one standard
  deviation below and above the mean of 128, which
  means that there is a 68 chance that the number
  of Heads will fall somewhere between 120 and 136.
  
• We should indeed make this bet!

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Normal Distributions

• The Dishonest-Coin Principle
• Suppose an arbitrary coin is tossed n times (n ?
  30), and let X denote the number of Heads that
  come up. Suppose also that p is the probability
  of the coin landing heads, and (1 p) is the
  probability of the coin landing tails. Then the
  random variable X has an approximately normal
  distribution with mean ? n p Heads and
  standard deviation
  Heads. .

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Normal Distributions

• Coin-Tossing Experiments Part 3
• Let p 0.20 n 100
• What can we say about X?
• ? n p 100 ? 0.20 20
• 4

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Normal Distributions

• Coin-Tossing Experiments Part 3
• Applying the 68-95-99.7 rule gives the
  following
• There is about a 68 chance that X will be
  somewhere between 16 and 24.
• There is about a 95 chance that X will be
  somewhere between 12 and 28.
• The number of Heads is almost guaranteed (about
  99.7 chance) to fall somewhere between 8 and 32.

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Normal Distributions Conclusion

• Bell-shaped (normal) curves
• Statistical Inference
• Laws of probability