2.2: Normal Distributions

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2.2 Normal Distributions

• Note on Uniform Distributions?

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3 Reasons why we like Normal Distributions

• Good descriptions of real data (ex SATs,
  psychological tests, characteristics of
  populations)
• Good approximations to results of many kinds of
  chance outcomes.
• Many inference procedures work well for roughly
  symmetrical distributions.
• Many data sets tend to be mound-shaped
  (characteristics of biological populations)
• TI83 student heights, L1, graph

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Normal Distributions

• Described by giving its mean and std.
  deviation
• controls the spread of a normal curve. Figure
  shows curve w/different values of .
• Changing w/o changing moves the
  curve along the horizontal axis w/o changing
  spread.

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Locating the standard deviation by eyeballing the
curve Inflection Points
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Common Properties of Normal Curves

• They all have inflection points (where change of
  curvature takes place).
• E. rule only provides an approximate value for
  the proportion of observations that fall within
  1, 2, or 3 std. devs of the mean.

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Example 1

• Suppose that taxicabs in NYC are driven an
  average of 75,000 miles per year with a standard
  deviation of 12,000 miles. What information does
  the empirical rule tell us?

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• 2 Normal curves
• What do you notice about their means?
• What do you notice about their standard
  deviations?

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Finding Areas to the Left

• Find the proportion of observations from the
  standard normal distribution that are less than
  2.22.
• That is
• Find P (z lt 2.22)

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Finding Areas to the Right

• Find the proportion of observations from the
  standard normal distribution that are greater
  than -2.15.
• That is find
• P (z gt -2.15)

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Table A Practice

• Use Table A to find the proportion of
  observations from a standard Normal distribution
  that falls in each of the following regions. In
  each case, sketch a standard Normal curve and
  shade the area representing the region.
• Z is less than or equal to -2.25
• Z is greater than or equal to -2.25
• Z gt 1.77
• -2.25 lt z lt 1.77

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Example

• The mean of women is 64.5 inches, and the
  standard deviation is 2.5 inches. What proportion
  of all young women are less than 68 inches tall?

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Example

• The level of cholesterol in the blood is
  important because high cholesterol levels may
  increase the risk of heart disease. The
  distribution of blood cholesterol levels in a
  large population of people of the same age and
  sex is roughly normal. For 14 year old boys, the
  mean is 170 mg/dl and the standard deviation is
  30 mg/ . Levels above 240 mg/dl may require
  medical attention. What percent of 14-year-old
  boys have more than 240 mg/dl of cholesterol?

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What percent of 14 year old boys have between 170
and 240 mg/dl?
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Finding a value given a proportion

• Use Table A backwards!
• 1) Find the given proportion in the body of the
  table
• 2) Read the corresponding z-score
• 3) Unstandardize to get the observed (x) value.
  Voila!

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Example

• Scores on the SAT verbal test in recent years
  follow approximately the N(505,110) distribution.
  How high must a student score in order to place
  in the top 10 of all students taking the SAT?

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Special Note.

• X is greater than is the same as x is greater
  than or equal to.
• That is, there is no area under the curve where x
  240. There may be a boy with an exact
  cholesterol level of 240, but there is no area
  under the curve at an exact point.
• The normal distribution is therefore an
  approximation not a description of every detail
  in the exact data.

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Normal Probability Plot

• TI83
• Dont overreact to minor wiggles in the plot
• Normality cannot be assumed if there is skewness
  or outliers (dont use Normal distribution if
  these things occur)!

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