The Normal Distributions

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Chapter 3

• The Normal Distributions

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• Z-Score Explained
• http//www.youtube.com/watch?vAT-HH0W_swAfeature
  related
• Basics of Using the Std Normal Table
• http//www.youtube.com/watch?vy6sbghmHwQAfeature
  related
• Normal Distribution Z-score
• http//www.youtube.com/watch?vmai23vW8uFMfeature
  related

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Well Learn The Topics

• Review Histogram
• Density Curve
• Normal Distribution
• 68 95 99.7 Rule
• Z-score
• Standard Normal Distribution

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Density Curves

• Example here is a histogram of vocabulary
  scores of 947 seventh graders.

• We can describe the histogram with a smooth
  curve, a bell- shaped curve.
• It corresponding to a normal distribution Model.

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Density Curves

• Example the areas of the shaded bars in this
  histogram represent the proportion of scores that
  are less than or equal to 6.0. This proportion
  in the observed data is equal to 0.303.

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Density Curves

• now the area under the smooth curve to the left
  of 6.0 is shaded.
• The scale is adjusted, the total area under the
  curve is exactly 1, this curve is called a
  density curve.
• The proportion of the area to the left of 6.0
  is now equal to 0.293.

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Density Curves

• Always on or above the horizontal axis
• Have area exactly 1 underneath curve
• Display the bell-shaped pattern of a distribution
• A histogram becomes a density curve if the scale
  is adjusted so that the total area of the bars is
  1.

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Mean Standard Deviation

• The mean and standard deviation computed from
  actual observations (data) are denoted by and
  s, respectively

• The mean and standard deviation of the
  distribution represented by the density curve are
  denoted by µ (mu) and ? (sigma),
  respectively.
• The mean of a density curve is the "balance
  point" of the curve.

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Bell-Shaped CurveThe Normal Distribution
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The Normal Distribution

• Knowing the mean (µ) and standard deviation (?)
  allows us to make various conclusions about
  Normal distributions.
• Notation N(µ,?).

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68-95-99.7 Rule forAny Normal Curve

• 68 of the observations fall within one standard
  deviation of the mean
• 95 of the observations fall within two standard
  deviations of the mean
• 99.7 of the observations fall within three
  standard deviations of the mean

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68-95-99.7 Rule forAny Normal Curve
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68-95-99.7 Rule forAny Normal Curve
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Health and Nutrition Examination Study of
1976-1980

• Heights of adult men, aged 18-24
• mean 70.0 inches
• standard deviation 2.8 inches
• heights follow a normal distribution, so we have
  that heights of men are N(70, 2.8).

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Health and Nutrition Examination Study of
1976-1980

• 68-95-99.7 Rule for mens heights
• 68 are between 67.2 and 72.8 inches
• µ ? ? 70.0 ? 2.8
• 95 are between 64.4 and 75.6 inches
• µ ? 2? 70.0 ? 2(2.8) 70.0 ? 5.6
• 99.7 are between 61.6 and 78.4 inches
• µ ? 3? 70.0 ? 3(2.8) 70.0 ? 8.4

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Health and Nutrition Examination Study of
1976-1980

• What proportion of men are less than 72.8 inches
  tall?

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Standard Normal Distribution

• Z Score
• The standard Normal distribution N(0,1) is the
  Normal distribution has a mean of zero and a
  standard deviation of one
• Normal distributions can be transformed to
  standard normal distributions by Z-score

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Health and Nutrition Examination Study of
1976-1980

• What proportion of men are less than 68 inches
  tall?

How many standard deviations is 68 from 70?
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Standardized Scores

• standardized score (Z-score)
• (observed value minus mean) / (std dev)
• (68 - 70) / 2.8 -0.71
• The value 68 is 0.71 standard deviations below
  the mean 70.

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Health and Nutrition Examination Study of
1976-1980

• What proportion of men are less than 68 inches
  tall?

-0.71 0 (standardized values)
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Table AStandard Normal Probabilities

• See pages 464-465 in text for Table A. (the
  Standard Normal Table)
• Look up the closest standardized score (z) in the
  table.
• Find the probability (area) to the left of the
  standardized score.

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Table AStandard Normal Probabilities
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Table AStandard Normal Probabilities
z .00 .02
?0.8 .2119 .2090 .2061
.2420 .2358
?0.6 .2743 .2709 .2676
.01
?0.7
.2389
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Health and Nutrition Examination Study of
1976-1980

• What proportion of men are less than 68 inches
  tall?

.2389
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Health and Nutrition Examination Study of
1976-1980

• What proportion of men are greater than 68
  inches tall?

1?.2389 .7611
.2389
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Health and Nutrition Examination Study of
1976-1980

• How tall must a man be to place in the lower 10
  for men aged 18 to 24?

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Table AStandard Normal Probabilities

• See pages 464-465 in text for Table A.
• Look up the closest probability (to .10 here) in
  the table.
• Find the corresponding standardized score.
• The value you seek is that many standard
  deviations from the mean.

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Table AStandard Normal Probabilities
z .07 .09
?1.3 .0853 .0838 .0823
.1020 .0985
?1.1 .1210 .1190 .1170
.08
?1.2
.1003
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Health and Nutrition Examination Study of
1976-1980

• How tall must a man be to place in the lower 10
  for men aged 18 to 24?

-1.28 0 (standardized values)
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Observed Value for a Standardized Score

• Need to reverse the z-score to find the
  observed value (x)

• observed value
• mean plus (standardized score) ? (std dev)

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Observed Value for a Standardized Score

• observed value
• mean plus (standardized score) ? (std dev)
• 70 (-1.28 ) ? (2.8)
• 70 (?3.58) 66.42
• A man would have to be approximately 66.42 inches
  tall or less to place in the lower 10 of all men
  in the population.

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The Entry in Table A

• Using random variable z to get the entrance in
  Table A.
• Variable z is z-score which follows the standard
  normal distribution N(0, 1)
• Z-score
• When search entry for a z value
• ? look up the most left column first, locate the
  most close value to z value
• ? look up the top row to locate the 2th decimal
  place for a z value

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The Entry in Table A

• Table As entry is an area underneath the curve,
  to the left of z
• Table As entry is a percent of the whole area,
  to the left of z-score
• Table As entry is a probability, corresponding
  to the z-score value.
• Math formula
• P (z z0) 0.xxxx
• P (z -0.71) 0.2389

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Problem type I

• If z N(0, 1), P (z z0) ?
• By checking the Table A, find out the answer.
• For type I problem, check the table and get the
  answer directly.
• For example,
• P (z -0.71) 0. 2389

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Problem type II

• If z N(0, 1), P (z z0) ?
• This types problem, cannot check the table
  directly. Using the following operation.
• P (z z0) 1 - (z z0)
• For example, p (z -0.71) ?
• ? P (z -0.71) 1 - (z - 0.71) 1 0.2389
  0.7611

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Problem Type II
0.7611
0.2389
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Problem Type III

• If z N(0, 1), P ( z2 z z1) ?
• random variable z is between two numbers
• Look up z1 ? P1
• Look up z2 ? P2
• The result is P ( z2 z z1) P1 - P2
• For example, P ( -1.4 z 1.3) ?
• look up 1.3 P1 0.9032
• look up -1.4 P2 0.0808
• P ( -1.4 z 1.3) 0.9032 0.0808 0.8224

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Steps Summary

• Write down the normal distribution N(µ,?) for
  observation data set
• Locate the specific observation value X0
• Transform X0 to be Z0 by z-score formula
• Check table A using random variable Z0 to find
  out table entry P(z z0)
• If is problem type I, the result is P(z z0)
• If is problem type II, the result is
• P (z z0) 1- P(z z0)
• If is problem type III, the result is
• P ( z2 z z1) P1 - P2
• P1 P(z z1), P2 P(z z2)