The Normal Distributions

1
Chapter 3

• The Normal Distributions

2
Density Curves

• Here is a histogram of vocabulary scores of n
  947 seventh graders
• The smooth curve drawn over the histogram is a
  mathematical model which represents the density
  function of the distribution

3
Density Curves

• The shaded bars on this histogram corresponds to
  the scores that are less than 6.0
• This area represents is 30.3 of the total area
  of the histogram and is equal to the percentage
  in that range

4
Area Under the Curve (AUC)

• This figure shades area under the curve (AUC)
  corresponding to scores less than 6
• This also corresponds to the proportion in that
  range AUC proportion in that range

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Density Curves
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Mean and Median of Density Curve
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Normal Density Curves

• Normal density curves are a family of bell-shaped
  curves
• The mean of the density is denoted µ (mu)
• The standard deviation is denoted s (sigma)

8
The Normal Distribution

• Mean µ defines the center of the curve
• Standard deviation s defines the spread
• Notation is N(µ,?).

9
Practice Drawing Curves!

• The Normal curve is symmetrical around µ
• It has infections (blue arrows) at µ s

10
The 68-95-99.7 Rule

• 68 of AUC within µ 1s
• 95 fall within µ 2s
• 99.7 within µ 3s
• Memorize!

This rule applies only to Normal curves
11
Application of 68-95-99.7 rule

• Male height has a Normal distribution with µ
  70.0 inches and s 2.8 inches
• Notation Let X male height X N(µ 70, s
  2.8)
• 68-95-99.7 rule
• 68 in µ ? ? 70.0 ? 2.8 67.2 to 72.8
• 95 in µ ? 2? 70.0 ? 2(2.8) 64.4 to 75.6
• 99.7 in µ ? 3? 70.0 ? 3(2.8) 61.6 to 78.4

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Application 68-95-99.7 Rule

• What proportion of men are less than 72.8 inches
  tall?
• µ s 70 2.8 72.8 (i.e., 72.8 is one s
  above µ)

Therefore, 84 of men are less than 72.8 tall.
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Finding Normal proportions

• What proportion of men are less than 68 tall?
  This is equal to the AUC to the left of 68 on
  XN(70,2.8)

To answer this question, first determine the
z-score for a value of 68 from XN(70,2.8)
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Z score

• The z-score tells you how many standard deviation
  the value falls below (negative z score) or above
  (positive z score) mean µ
• The z-score of 68 when XN(70,2.8) is

Thus, 68 is 0.71 standard deviations below µ.
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Example z score and associate value
-0.71 0 (z values)
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Standard Normal Table
Use Table A to determine the cumulative
proportion associated with the z score
See pp. 79 83 in your text!
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Normal Cumulative Proportions (Table A)
z .00 .02
?0.8 .2119 .2090 .2061
.2420 .2358
?0.6 .2743 .2709 .2676
.01
?0.7
.2389
Thus, a z score of -0.71 has a cumulative
proportion of .2389
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Normal proportions
The proportion of mean less than 68 tall
(z-score -0.71 is .2389
.2389
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Area to the right (greater than)
Since the total AUC 1 AUC to the right 1
AUC to left Example What of men are greater
than 68 tall?
1?.2389 .7611
.2389
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Normal proportions
The key to calculating Normal proportions is to
match the area you want with the areas that
represent cumulative proportions. If you make a
sketch of the area you want, you will almost
never go wrong. Find areas for cumulative
proportions from Table A (p. 79) Follow the
method in the picture (see pp. 79 80) to
determine areas in right tails and between two
points
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Finding Normal values

• We just covered finding proportions for Normal
  variables. At other times, we may know the
  proportion and need to find the Normal value.
• Method for finding a Normal value
• 1. State the problem
• 2. Sketch the curve
• 3. Use Table A to look up the proportion
  z-score
• 4. Unstandardize the z-score with this formula

22
State the Problem Sketch Curve
Problem How tall must a man be to be taller than
10 of men in the population? (This is the same
as asking how tall he has to be to be shorter
than 90 of men.) Recall XN(70, 2.8)
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Table AFind z score for cumulative proportion
.10
z .07 .09
?1.3 .0853 .0838 .0823
.1020 .0985
?1.1 .1210 .1190 .1170
.08
?1.2
.1003
zcum_proportion z.1003 -1.28
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Visual Relationship Between Cumulative proportion
and z-score
-1.28 0 (Z value)
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Unstandardize

• x µ zs 70 (-1.28 )(2.8) 70
  (?3.58) 66.42
• Conclude A man would have to be less than 66.42
  inches tall to place him in the lowest 10 of
  heights