The normal distribution

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The normal distribution
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Outline

• Describing data with a smooth curve
• The normal curve
• The 68-95-99.7 rule
• The standard normal distribution
• Calculating the area under the normal curve
• The standard normal table
• Computations for non-standard normal distributions

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• When examining data on a single quantitative
  variable
• Plot the data
• Look for overall pattern
• Calculate numerical summaries to briefly describe
  center and spread
• When the overall pattern is quite regular, use a
  compact mathematical model to describe it.

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Describing data with a smooth curve
Survival times of 72 guinea pigs injected with
tubercle bacilli Smooth curve is estimated
using software No easy math formula for this
curve
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Distribution of blood pressure The histogram
shows 1,000 blood pressures
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Vocabulary scores of 947 seventh-grade students
in Gary, Indiana.
Survey 0200 spring 2005 morning.MPJ Survey 0200
spring 2005 evening.MPJ
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The normal curve(The Normal (Gaussian)
distribution)

• Normal curve has a particular symmetric, bell-
• shaped pattern, which can be summarized with this
• equation

Notation
e.g., SAT scores N(505,1102)
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Changing µ and s
Changing µ changes the center Changing s changes
the spread
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Calculating the area under the normal curve
The 68-95-99.7 rule
f(x)
-3s -2s -s µ s
2s 3s
68 of observations fall within s of the mean µ.
95 of observations fall within 2s of the mean
µ. 99.7 of observations fall within 3s of the
mean µ.
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The standard normal distribution

• Normal variables may have different µ and s

We can standardize each normal variable to have
µ0 and s1 If the variable X has a normal
distribution N(µ,s) then
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Example

• Womens height is N(µ64.5,s2.5)
• Marys height is 67. Her standardized height is
• Marys height is ___ standard deviation above the
  mean

1
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Example

• Reading ability of third grade children is
  N(µ75, s10).
• Reading ability of sixth grade children is
  N(µ82, s11).
• David, 3rd grade, scored 80.
• Nancy, 6th grade scores 80.
• Who scored better relative to the their grade?
• Davids standard score is
• (80-75)/100.5
• Nancys standard score is
• (80-82)/11-0.18
• Conclusion
• Relative to their grades, David scored higher
  than Nancy

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Calculating the area under the normal curve

• Difficulty
• Integrating over the formula of the N(µ,s) Curve
  involves difficult numerical computations.
• The solution
• we standardize the N(µ, s) to have common µ and
  s ? µ0 and s1.
• There are tables that already performed these
  computations for the N(0,1)

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Using the Normal table to compute area under the
Normal curve
Table entry
z
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Examples of using the Normal table
1. Z -1.5
F(-1.5) 0.0668 6.7
-1.5
F(-1.51) 0.0655 6.6
2. Z -1.51
-1.51
1- F(1.5) 1-.9332 0.0668 6.7
3. Z1.5
1.5
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F(1.5)-F(-1.5)0.9332-0.0668 0.8664 86.6

• 5. -2Z-1

4. -1.5Z1.5
1.5
-1.5
?
-2 -1
F(-1)-F(-2) 0.1587-0.0228 0.1357 13.6
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• Finding Z for a given area
• 25 of normal curve lies to the left of what
  value?
• 15 lies to the right of what value?
• 85 lies to the right of what value?

25
Z0.25 -0.675
Z___
15
Z0.85 1.035
Z___
85
Z0.15 -1.035
Z___
Go to Practice
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More practice
Find the shaded area
F(-1.65)0.0495
-1.65
1-F(2.15)1-0.98420.0158
2.15
0.2
Z0.2-0.845
z?
0.35
Z0.650.385
z?
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The normal table with Minitab
Calc gt Probability distributions gt Normal
Minitab MTBWIN\Mtb13.exe
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Computations for a non-standard normal
distribution
Assume that

• What of people have IQ between 85 and 115?
• To use the normal table we have to transform to
  standard units

?
85
115
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• Convert 85 and 115 to standard units

68
____
85 100 115
-1 0 1
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• What of people have IQ below 95?

95 100
Z____ 0
Answer
37.07
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• What of people have IQ below 80?

80 100
Z___ 0
Convert 80 to standard units
Find area using normal table
Answer 9.18
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• What of people have IQ above 140?

0 Z____
100 140
Convert 140 to standard units
Area is
Answer
0.38
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• What of people have IQ between 100 and 120?

100 120
Z___ Z____
Convert 120 to standard units
Subtract the area below 0 from the area below
1.33 Area below 0 0.5 Area below 1.33 is
0.9082 0.9082-0.50.4082
Answer 40.8
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• What is the min score that someone must achieve
  to be in
• the top 5?

5
100 x
?
First, find what standard score is appropriate
Find IQ value that is 1.645 standard deviations
above the mean
X124.675
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Practice non-standard normal distribution

• Scores on the SAT verbal test follow
  approximately
• N(µ505,s110) distribution.
• 1. What percent of students have scores higher
  than 700?

?
700
Transform 700 into standard units
(700-505)/1101.77
1-0.96160.0384
1.77
Answer 3.84
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• 2. What percent of students have scores between
  505 and 700?

?
505 700
Transform to standard scores
(700-505)/1101.77 (505-505)/1100
?
F(1.77)-F(0)0.9616-0.50.4616
0 1.77
Answer is 46.16
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• 3. How high must a student score in order to
  place in the top 10 of all students taking the
  SAT?

10
X
First find the standard score that is below the
top 10
Find SAT score that is equivalent to 1.285 z score
X646.35
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• 4. Suppose that in order to pass the verbal part
  of the SAT you need a score which is not within
  the lowest 10. What SAT score do you need to get
  in order to pass the verbal part of the SAT?

10
X
First find the standard score that is above the
top 10
Find SAT score that is equivalent to 1.285 z score
X363.65
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Solve the SAT Question using Minitab Minitab
The normal table