The Normal Distribution

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Chapter 6

• The Normal Distribution

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Normal Distributions

• Bell Curve
• Area under entire curve 1 or 100
• Mean Median
• This means the curve is symmetric

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Normal Distributions

• Two parameters
• Mean µ (pronounced meeoo)
• Locates center of curve
• Splits curve in half
• Shifts curve along x-axis
• Standard deviation s (pronounced sigma)
• Controls spread of curve
• Smaller s makes graph tall and skinny
• Larger s makes graph flat and wide
• Ruler of distribution
• Write as N(µ,s)

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Standard Normal (Z) World

• Perfectly symmetric
• Centered at zero
• Half numbers below the mean and half above.
• Total area under the curve is 1.
• Can fill in as percentages across the curve.

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Standard Normal Distribution

• Puts all normal distributions on same scale
• z has center (mean) at 0
• z has spread (standard deviation) of 1

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Standard Normal Distribution

• z of standard deviations away from mean µ
• Negative z, number is below the mean
• Positive z, number is above the mean
• Written as N(0,1)

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Portal from X-world to Z-world

• z has no units (just a number)
• Puts variables on same scale
• Center (mean) at 0
• Spread (standard deviation) of 1
• Does not change shape of distribution

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Standardizing Variables

• z of standard deviations away from mean
• Negative z number is below mean
• Positive z number is above mean

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Standardizing

• Y N(70,3). Standardize y 68.
• y 68 is 0.67 standard deviations below the mean

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Your Table is Your Friend

• Get out your book and find your Z-table.
• Look for a legend at the top of the table.
• Which way does it fill from?
• Find the Z values.
• Find the middle of the table.
• These are the areas or probabilities as you move
  across the table.
• Notice they are 50 in the middle and 100 at the
  end.

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Areas under curve

• Another way to find probabilities when values are
  not exactly 1, 2, or 3 ? away from µ is by using
  the Normal Values Table
• Gives amount of curve below a particular value of
  z
• z values range from 3.99 to 3.99
• Row ones and tenths place for z
• Column hundredths place for z

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Finding Values

• What percent of a standard Normal curve is found
  in the region Z lt -1.50?
• P(Z lt 1.50)
• Find row 1.5
• Find column .00
• Value 0.0668

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Finding Values

• P(Z lt 1.98)
• Find row 1.9
• Find column .08
• Value 0.9761

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Finding values

• What percent of a std. Normal curve is found in
  the region Z gt-1.65?
• P(Z gt -1.65)
• Find row 1.6
• Find column .05
• Value from table 0.0495
• P(Z gt -1.65) 0.9505

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Finding values

• P(Z gt 0.73)
• Find row 0.7
• Find column .03
• Value from table 0.7673
• P(Z gt 0.73) 0.2327

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Finding values

• What percent of a std. Normal curve is found in
  the region 0.5 lt Z lt 1.4?
• P(0.5 lt Z lt 1.4)
• Table value 1.4 0.9192
• Table value 0.5 0.6915
• P(0.5 lt Z lt 1.4) 0.9192 0.6915 0.2277

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Finding values

• P(2.3 lt Z lt 0.05)
• Table value 0.05 0.4801
• Table value 2.3 0.0107
• P(2.3 lt Z lt 0.05) 0.4801 0.0107
  0.4694

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Finding values

• Above what z-value do the top 15 of all z-value
  lie, i.e. what value of z cuts offs the highest
  15?
• P(Z gt ?) 0.15
• P(Z lt ?) 0.85
• z 1.04

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Finding values

• Between what two z-values do the middle 80 of
  the obs lie, i.e. what values cut off the middle
  80?
• Find P(Z lt ?) 0.10
• Find P(Z lt ?) 0.90
• Must look inside the table
• P(Zlt-1.28) 0.10
• P(Zlt1.28) 0.90

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Solving Problems

• The height of men is known to be normally
  distributed with mean 70 and standard deviation
  3.
• Y N(70,3)

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Solving Problems

• What percent of men are shorter than 66 inches?
• P(Y lt 66) P(Zlt ) P(Zlt-1.33)
  0.0918

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Solving Problems

• What percent of men are taller than 74 inches?
• P(Y gt 74) 1-P(Ylt74) 1 P(Zlt
  ) 1 P(Zlt1.33) 1 0.9082
  0.0918

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Solving Problems

• What percent of men are between 68 and 71 inches
  tall?
• P(68 lt Y lt 71) P(Ylt71) P(Ylt68)P(Zlt
  )-P(Zlt )P(Zlt0.33) -
  P(Zlt-0.67) 0.6293 0.2514 0.3779

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Solving Problems

• Scores on SAT verbal are known to be normally
  distributed with mean 500 and standard deviation
  100.
• X N(500,100)

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Solving Problems

• Your score was 650 on the SAT verbal test. What
  percentage of people scored better?
• P(X gt 650) 1 P(Xlt650) 1 P(Zlt
  )
• 1 P(Zlt1.5) 1 0.9332
  0.0668

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Solving Problems

• To solve a problem where you are looking for
  y-values, you need to rearrange the standardizing
  formula

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Solving Problems

• What would you have to score to be in the top 5
  of people taking the SAT verbal?
• P(X gt ?) 0.05?
• P(X lt ?) 0.95?

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Solving Problems

• P(Z lt ?) 0.95?
• z 1.645
• x is 1.645 standard deviations above mean
• x is 1.645(100) 164.5 points above mean
• x 500 164.5 664.5
• SAT verbal score at least 670

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Solving Problems

• Between what two scores would the middle 50 of
  people taking the SAT verbal be?
• P(x1 ? lt X lt x2?) 0.50?
• P(-0.67 lt Z lt 0.67) 0.50
• x1 (-0.67)(100)500 433
• x2 (0.67)(100)500 567

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Solving Problems

• Cereal boxes are labeled 16 oz. The boxes are
  filled by a machine. The amount the machine
  fills is normally distributed with mean 16.3 oz
  and standard deviation 0.2 oz.

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Solving Problems

• What is the probability a box of cereal is
  underfilled?
• Underfilling means having less than 16 oz.
• P(Y lt 16) P(Zlt ) P(Zlt
  -1.5) 0.0668

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Moving to Random Sample

• Even when we assume a variable is normally
  distributed if we take a random sample we need to
  adjust our formula slightly.

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Determining Normality

• If you need to determine normality I want you to
  use your calculators to make a box plot and to
  look for symmetry.