Density Curves

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Density Curves Normal Distribution
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The left end point of each bin (except 1st) is
not included, the right end point is included in
that bin.
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Percentages Based On Histogram

• How many states have more than 8 Hispanic
  population?
• What proportion of states (/50) having between 4
  and 8 Hispanic population?
• What proportion of states having between 12 and
  20 Hispanic population?
• What proportion of states having between 11 and
  19 Hispanic population?
• Sum of all percentages 100 1.

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If you choose smaller bins
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Introducing Density Curve

• An overall shape total area below which is 1.
  This reflects the fact the total percentage is 1.
• It can tell us what percentage of the subjects
  lies between any two numbers, computed by area.
  For a histogram that uses bins 0,4),4,8), etc,
  it cant tell us how many or what percentage of
  subjects lies between 2 and 5 or between 11 and
  19.
• For comparison, the percentage or the frequency
  is given by the height of the bin of a histogram.

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Uniform distribution
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Uniform distribution
Uniform distribution
Normal distribution bell-shaped, symmetric and
extend in both directions
Normal distribution bell-shaped, symmetric and
extend in both directions
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Mean and Median Based On Density Curve

• 50 of the data lies below the median, 50 of
  the data lies above the median. We use area to
  indicate percentage, so the median of a density
  curve is the point that splits the total area
  under the curve in half. Each has area 0.5.
• 1 student has 40, 4 others have 15 each. The
  mean is 20 the pole-balancing point. The mean
  of a density curve is the balance point, at which
  the curve would balance if made of evenly
  produced solid material.
• For a symmetric density curve, mean median
  the center of the curve The mean of a skewed
  density curve is pulled away from the median in
  the direction of the long tail.
• Examples on previous slides.

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What are the mean and standard deviation of a
normal distribution?
The notation for this normal distribution
2.5
-3
0
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Normal Distributions

• A normal distribution is a very special type of
  distribution whose density curve is symmetric and
  bell-shaped.
• There are many different normal distributions,
  depending on the mean m and standard deviation s.
• The mean m the median the horizontal middle
  point of the density curve. The standard
  deviation s is the distance of turning point
  (think of downhill skiing) away from the middle.
• The mean m and the standard deviation s
  completely determine the normal distribution, so
  we write normal distribution N(m , s ).
• Normal distributions tails are infinite in both
  directions.

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How does the normal distribution N (0,1) look
like?
N (-1, 2.3)
N (123, 1)
1
1
0
123
The normal distribution N (0, 1), with mean 0,
standard deviation 1, is called
Exercise 3.5, p 64
The Standard Normal Distribution
Exercise 3.7, p 65
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Normal Distribution Examples and Calculations

• Example 1 The length of human pregnancies is
  approximately normal N (266, 16).
• Questions (draw pictures to interpret!)

• What percentage of pregnancies last less than
  266 days? 260 days? 280 days?
• What percentage of pregnancies last between 260
  days and 280 days?
• What percentage of pregnancies last more than
  275 days?
• Between what values do the lengths of the middle
  90 of all pregnancies fall?
• How short are the shortest 25 of all
  pregnancies?
• What are the Q1 and Q3 of this distribution?

• Example 2 Example 3.6 p 69, Example 3.7 p 70

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Normal Distribution Calculations

• Using normal distribution Java applet
• Computing z-score and Table A lookup
• 68-95-99.7 rule
• Using Excel

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Using Java Applet

• Choose the normal curve applet from the course
  website.
• Or go to http//statweb.calpoly.edu/chance/applets
  /applets.html.
• And choose Normal Probability Calculations.
• Exercises 3.113.13, p 71 and p 73

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Z-score

• The heights of women aged between 20 to 29 are
  approximately Normal with mean 64 inches and
  standard deviation 2.7 inches.
• m 64 inches, s 2.7 inches.
• If Joans height is 61.3 inches, whats the
  deviation from the mean height?
• How many standard deviations is Joans height
  from the mean height?
• If Norahs height is 69.4 inches, answer the
  above two questions.
• If Samanthas height is 62 inches, answer the
  above two questions.

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Z-score

• Given a normal distribution N (m, s) and an
  observation x, how many standard deviations is x
  from the mean m?
• The answer is called the z-score or standardized
  score of x!
• The z-score of x measures how many standard
  deviations x is from the mean.
• The relation of x and its z-score z is given in
  the following formulae

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Z-score

• A Health and Examination study found the heights
  of men aged between 2029 follow the normal
  distribution with mean 70 inches, standard
  deviation 2.8 inches.
• Whats the z-score of Tims height 80 inches?
  Peters height 65 inches?
• Which height has the z-score equal to -2? 3.1? 0?
• Positive z-score indicates height above the mean
  negative z-score indicates height below the mean.

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Table A Lookup

• In N(70,2.8), what percentage of mens height is
  below 68? The z-score of 68 is (68-70)/2.8 -0.71
  (limited two-decimal places in table A).
• What percentage of mens height is below 73? The
  z-score of 73 is (73-70)/2.81.07.

Note 1 drawing a picture helps you understand
these numbers .
Note 2 given z, the corresponding number in the
table A is always the area to the left of x
under the normal density curve.
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Mens Heights
Standardized score
( two decimal places)
Answer 23.89 of men are less than 68 inches tall
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Mens Heights
?
Standardized score
Answer1-0.8577 0.1423 14.23 of men are
taller than 73 inches
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Mens Heights
?
Standardized score
Relationship percentage between percentage
below 73 percentage below 68
Answer 0.8577-.2389 0.6188 61.88 of men are
taller than 68 inches and shorter than 73 inches
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Mens Heights
? At most how tall can a man be if he is among
the shortest 10 (among the subjects in the
study).
Standardized score
Answer
Analysis If we know ? ? compute its z-score ?
look up table A and find the area to the left of
?. We now know the area to the left of ? 10!
? Table A lookup to find z-score (the number of
standard deviations of ? from the mean height 70)
? find ?
Reverse lookup find 10 0.1 in the table,
whats z? Closest number to 0.1 is 0.1003. The
corresponding z is 1.28. So he must be taller
than 1.28 standard deviations from the mean to
avoid being in the lowest 10.
Example 3.8 p 72 Exercise 3.13 p 73.
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The Standard Normal Distribution N(0,1)

• A normal distribution whose standard deviation is
  1 and whose mean is 0, is called the standard
  normal distribution. Notation N(0,1).
• In N(0,1) 3 is -3 standard deviations from the
  mean 0 2 is 2 standard deviations from the mean.
  The z-score of x is x.
• Many normal distributions N(µ,?) depending on µ
  and ? , but only one standard normal
  distribution.
• If a variable x has any normal distribution with
  mean µ and standard deviation ? x N(µ,?) ,
  then the following standardized variable
  (standardized score) has the standard normal
  distribution

Exercises 3.183.19, p 75
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68-95-99.7 Rule

• Whats the area under the density curve of N(µ,?)
  between a µ- ? (one standard deviation to the
  left of µ) and b µ? (one standard deviation to
  the right of µ)?

Answer 0.68

• The area under the density curve of N(µ,?)
  between µ- 2? (two standard deviations to the
  left of µ) and µ2? (two standard deviations to
  the right of µ) is 0.95 the area under the
  density curve of N(µ,?) between µ- 3? and µ3? is
  0.997.

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68-95-99.7 Rule
2 standard deviations from m
Mens Heights N (70, 2.8)

• 68 are between 67.2 and 72.8 inches
• µ ? ? 70.0 ? 2.8 between one s left and
  one s right
• 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

Exercises 3.63.7, p 64
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Mens Heights N (70, 2.8)

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

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Excel Instructions Normal Distributions

• N( m, s ) (Example N(17,3) )
• The area to the left of x is Enter
• normdist(x, m, s, 1) (N(13,17,3,1))
• The area between a and b (altb) is Enter
• normdist(b, m, s, 1)-normdist(a, m, s,
  1)
• (normdist(15,17,3,1)-normdist(13,17,3,1))
• The area to the right of x is Enter
• 1-normdist(x, m, s, 1)
  (1-normdist(19,17,3,1))
• The observation to the left of which the area is
  p is Enter
• norminv(p, m, s) (norminv(0.4,17,3))
• For Standard Normal Distribution, you can use
  normsdist and normsinv instead of normdist and
  norminv

No need to worry about z-score When using Excel.
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Time taken by new typists to learn a new computer
typing system is N(90,18)
(in minutes)
?
Use three methods (Java applet, z-score and table
A lookup, excel) to solve the following questions.
What proportion of typists need between 63
minutes and 100 minutes?
Answer .7123-.0668 .6455 64.55
In order not to be in the slowest 15, how fast
should a typist have to learn the system?
Answer should spend less than 108.72 minutes
learning the system
Use 68-95-99.7 rule to answer what percentage of
typists take longer than 126 minutes to learn the
new typing system?
Answer 2.5