Download 22-the-normal-distribution

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 2:
Describing Location in a Distribution
Section 2.2
Normal Distributions
Copyright © 2008 by W. H. Freeman & Company
Essential Questions
• What are the properties of the Normal
Curve?
• How do you explain the 68-95-99.7
empirical rule?
• What does N(µ, σ) mean?
• What is the standard Normal distribution?
The mathematical model for the
Normal Density Curve
 1 
f ( x)  
e
  2 
1  x 
 

2  
2
Why the Normal Distribution?
Many measurements distributions in the real world can be
modeled by the Normal Distribution
• Scores on Test (SAT, ACT, psychological
test, etc.)
• Repeated careful measurements of the
same quantity
• Characteristics of biological population
Beware- many real world distributions many not
follow the nomal distributions.
The Normal Curve
Point of
Inflection
Normal curves are
defined by the
means and the
standard
deviation.
Short Notation for
Normal Density Curves
Since the normal density curves are completely defined by the means μ
and the standard deviation σ, we can use the abbreviation:
N(μ, σ)
Characteristics of a Normal Density
Curve
• Symmetric
• Single peaked – bell shaped
• Means and median are located at the center of
symmetry for the curve
• The standard deviation is the distance from the
Means to the point of inflection on the curve
• The curve is described by means, μ, and
standard deviation, σ
• There are many normal curves
• All curves follow the 68-95-99.7 rule
The 68-95-99.7 Rule
The Normal Curve Applet
• Turn to page 137 Activity 2C
• Export-9-4-2008\NormalCurve.html
68-95-99.7 Rule
68.2-95.4-99.7 Rule
We will use the simplified version in the book: 68-95-99.7 rule.
68-95-99.7 Percentile
• Percentile score is the percent of individuals who scored less than or
equal to your score.
• Percentile is used for comparing an individual observation relative to
other individuals in the distribution.
• In practice, when observations are quite large then the percentile is
reported using density curves.
Percentile Using Density Curve
0.15% 2.5%
16%
50%
84%
97.5% 99.85%
Example
The distribution of heights for women is N(64.5”, 2.5”).
•
μ = ____
σ = _____
• The middle 68% of women are between
_______________
• The middle 95% are between ________
• The 50th percentile of women are ____ or lower.
• The 97.5th percentile of women are ____ or lower.
• ___% of women are 67” or shorter.
• ___% of women are 59.5” or taller.
• The ____ percentile would be women about 71.5” or
shorter.
The Standard Normal Curve
Since all normal density curves, N(μ, σ)
have the same characteristics we can use
one curve to do all of our calculations.
This curve is the Standard Normal Curve
N(0,1).
μ = 0 and σ = 1
When we change any N(µ, σ) to N(0, 1) we can find areas for all
N(µ, σ) using one single table called a Standard Normal Table.
• Turn to Table A in front of the Textbook.
• Notice: one side is for negative z-scores and one side for
positive z’s.
Example Using the Standard
Normal Table A
Find the proportion of the data from the standard
Normal distribution that will have a z-score of less than
-2.15.
Standard Normal Calculations
Export-9-4-2008\NormalCurve.html
• Find the proportion of observations from a
standard Normal distribution that falls in
each of the following region.
Ans: 0.0122
• z ≤ -2.25
• z ≥ -2.25
Ans: 1 – 0.0122 = 0.9878
• z > 1.77
Ans: 1 – 0.9616 = 0.0384
• -2.25 < z < 1.77 Ans: 1 – ( 0.0384 + 0.0122) =
0.9494
Working in Reverse
• Use Table A to find the value z of a standard Normal
variable that satisfies each of the following conditions. In
each case sketch a standard Normal curve with your
value of z marked on the axis.
• The point z with 70% of the observations falling below it.
• Ans: z = 0.52.
• The point with 85% of the observations falling above it.
• Ans: z = -1.04.
• Find the number z such that the proportion of
observations are less than z is 0.8.
• Ans: z = 0.84.
• Find the number z such that 90% of all observations are
greater than z.
• Ans: z = -1.28.
Procedure for Normal Distribution
Calculations
Example
During World War II, physical training was required for male students in many
colleges, as preparation for military service. That provided an opportunity to
collect data on physical performance on a large scale. A study of 12,000 ablebodied male students at the University of Illinois found that their times for the
mile run had a mean 7.11 minutes and a standard deviation of 0.739. The
distribution of times was roughly Normal. That seems pretty fast; presumably
the times were taken after the required training.
µ = 7.11 minutes
σ = 0.739 n = 12,000
(a) About how many of the 12,000 male students ran a mile in less than 5
minutes?
Using format in Exercise 1.17 on
page 65.
Example
N(7.11 min, 0.739)
a). About how many of the 12,000 male students ran a mile less
than 5 minutes?
Step 1 State the problem.
z
X < 5 minutes
Step 2 Standardize and draw.
5  7.11
z
 2.86
0.739
Step 3. Use Table A.
Proportion = 0.0021
Step 4 (0.0021)(12000) = 25.2
Approximately 25 students run less than 5 minutes per mile.
Example
• N(7.11, 0.739)
• What is the 93rd percentile in the mile run
time distribution?
z
Step 1 Proportion = .93
What is z?
Step 2 From table A z = 1.48
Step 3. Unstandardizing.
X  7.11
1.48 
0.739
X  (1.48)(0.739)  7.11  8.20 minutes
Step 4 Conclusion. The 93rd percentile is 8.20 minutes.
Checking for Normal
Two method
• Method I Construct a histogram or a stemplot and compare to the 68-95-99.7
empirical rule.
• Method 2 Construct a Normal probability
plot.
Method 1
• Construct a histogram or stem-plot.
• Locate x, x  s, x  2s, x  3s
On the x axis.
• Count the data that falls in each interval
and compare how well it compares to the
68-95-97.5 rule.
Manual Procedure for Constructing
a Normal Probability Plot
• Arrange the observed data in ascending
order.
• Change each observed data to a z-score
• Plot each data point x against the
corresponding z.
• If the data distribution is close to Normal
the plotted points will lie close to a straight
line.
Normal Probability Plots
Approximately Normal
Skewed Right distribution