Section 2.1 Density Curves

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Section 2.1Density Curves
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• Get out a coin and flip it 5 times. Count how
  many heads you get.
• Repeat this trial 10 times.
• Create a histogram for your data (frequency of
  how many heads you got in each of the 10 trials).
• Put your histogram on the board.

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Remember

• When we explore data on a single quantitative
  variable
• Plot your data (usually a histogram or stemplot)
• Look for the overall pattern (center, shape, and
  spread) and for outliers
• Calculate a numerical summary to describe center
  (median or mean) and spread (IQR or standard
  deviation)

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From Histograms to Curves

• Sometimes, the overall pattern from a large
  number of observations is so regular that we can
  overlay a smooth curve.

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Mathematical Models

• This curve is a mathematical model, or an
  idealized description, for the distribution.
• It is easier to work with the smooth curve than
  with the histogram.

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Density Curves

• Density curves are always positive (meaning its
  always on or above the horizontal axis).
• Areas under the curve represent proportions of
  the observations. The area under a density curve
  always equals 1.
• The density curve describes the overall pattern
  of a distribution. The area under the curve,
  within a range of values, is the proportion of
  all observations that fall in that range.

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Quartiles

• How much area would be to the left of the first
  quartile?
• How much area would be to the right of the first
  quartile?
• How much area would be between the first and
  third quartiles?

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What Does All of This Mean?

• When a density curve is a geometric shape
  (rectangle, trapezoid, or a combination of
  shapes) we can use geometry to find areas. Those
  areas help us find the median and the quartiles.

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• Verify that the graph is a valid density curve.
• For each of the following, use areas under
  density curve to find the proportion of
  observations within the given interval.
• 0.6ltXlt0.8
• 0ltXlt0.4
• 0ltXlt0.2

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• The median of this density curve is a point
  between X 0.2 and X 0.4. Explain why.

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Im seeing Greek!

• In a distribution, mean is x-bar and the standard
  deviation is s.
• When looking at a density curve, the mean is µ
  (pronounced mu) and the standard deviation is s
  (pronounced sigma).

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Normal Distributions

• Density curves have an area 1 and are always
  positive.
• Normal curves are a special type of density
  curves. Normal curves are symmetrical density
  curves.
• T/F All density curves are normal curves.
• T/F All normal curves are density curves.

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Characteristics of Normal Curves

• Symmetric
• Single-peaked (also called unimodal)
• Bell-shaped

µ
s
The mean, µ, is located at the center of the
curve. The standard deviation, s, is located at
the inflection points of the curve.
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Parameters of the Normal Curve

• A normal curve is defined by its mean and
  standard deviation.
• Notation for a normal curve is N(µ, s).

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Why Be Normal?

• Normal curves are good descriptions for lots of
  real data SAT test scores, IQ, heights, length
  of cockroaches (yum!).
• Normal curves approximate random experiments,
  like tossing a coin many times.
• Not all data is normal (or even approximately
  normal). Income data is skewed right.

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The Empirical Rulea.k.a. 68-95-99.7 Rule

• All normal distributions follow this rule
• 68 of the observations are within one standard
  deviation of the mean
• 95 of the observations are within two standard
  deviations of the mean
• 99.7 of the observations are within three
  standard deviations of the mean

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Yay, Math!

• IQ scores on the WISC-IV are normally
  distributed with a mean of 100 and a standard
  deviation of 15.
• Going up one s and down one s from 100 gives us
  the range from 85 to 115. 68 of people have an
  IQ between 85 and 115.
• 95 of people have an IQ between ____ and ____.
• 99.7 of people have an IQ between ____ and ____.

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Try This

• The heights of women aged 18 24 are
  approximately normally distributed with a mean µ
  64.5 inches and a standard deviation s 2.5
  inches.
• Between what two heights do the middle 95 fall?
• The tallest 2.5 of women are taller than what?
• What is the percentile for a woman who is 64.5
  inches tall?

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One more example

• The army reports that the distribution of head
  circumference among male soldiers is
  approximately normal with mean 22.8 inches and
  standard deviation 1.1 inches.
• What percent of soldiers have a head
  circumference greater than 23.9 inches?
• What percentile is this?
• What percent of soldiers have a head
  circumference between 21.7 inches and 23.9 inches?

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Homework

• Chapter 2 15a, 25, 41-45