Download Chapter 2 The Normal Distribution

The Normal Model
Chapter 6
Density Curves and
Normal Distributions
A Density Curve:
• Is always on or above the x axis
• Has an area of exactly 1 between the
curve and the x axis
• Describes the overall pattern of a
distribution
• The area under the curve above any range
of values is the proportion of all the
observations that fall in that range.
Mean vs Median
• The median of a density curve is the equal
area point that divides the area under the
curve in half
• The mean of a density function is the
center of mass, the point where curve
would balance if it were made of solid
material
Normal Curves
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Bell shaped, Symmetric,Single-peaked
Mean = µ

Standard deviation =
Notation N(µ,  )
One standard deviation on either side of µ
is the inflection points of the curve
68-95-99.7 Rule
• 68% of the data in a normal curve at least
is within one standard deviation of the
mean
• 95% of the data in a normal curve at least
is within two standard deviations of the
mean
• 99.7% of the data in a normal curve at
least is within three standard deviations of
the mean
Why are Normal Distributions
Important?
• Good descriptions for many distributions of
real data
• Good approximation to the results of many
chance outcomes
• Many statistical inference procedures are
based on normal distributions work well for
other roughly symmetric distributions
Standard Normal Curve
Chapter 6
Standardizing (z-score)
• If x is from a normal population with mean
equal to µ and standard deviation,  then
the standardized value z is the number of
standard deviations x is from the mean
• Z = (x - µ)/
• The unit on z is standard deviations
Standard Normal Distribution
• A normal distribution with µ = 0 and  = 1,
N(0,1) is called a Standard Normal
distribution
• Z-scores are standard normal where
z=(x-µ)/
Standard Normal Tables
• Table A in the front of your book has the areas to the left
of given cut-off points
• Find the 1st 2 digits of the z value in the left column and
move over to the column of the third digit and read off
the area.
• To find the cut-off point given the area, find the closest
value to the area ‘inside’ the chart. The row gives the
first 2 digits and the column give the last digit
Solving a Normal Proportion
• State the problem in terms of an x variable in the context
of the problem
• Draw a picture and locate the required area
• Standardize the variable using z =(x-µ)/
• Use the calculator/table and the fact that the total area
under the curve = 1 to find the desired area
Finding a Cutoff Given the Area
• State the problem in terms of x and area
• Draw a picture and shade the area
• Use the table to find the z value with the
desired area
• Unstandardize the z value using
z
=(x-µ)/ and solving for x
Assessing Normality
• In order to use the previous techniques the
population must be normal
• Method 1 for assessing normality :
 Construct a stem plot or histogram and see if
the curve is bell shaped and symmetric
around the mean
Assessing Normality: Revisited
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Normality Probably Plot on calculator
Enter data into a list
Stat Plot select graph type number 6
Specify the correct list and the data axis
as x
• Zoom Stat
• If the graph is nearly linear the distribution
is nearly normal
TI–83/84 Commands
Normal Curves
Chapter 6
Graphing Normal Curves
• Setting the window is NOT automatic. The
user (you) must set an appropriate window
• Window settings suggestions
 X[mean – 4 (Std dev), mean + 4(Std dev)]
 Xscale = Std dev
 Y[–.01, .02]
 Yscale = .01
ShadeNorm(xmin,xmax,mean,sd
)
• Draws the graph and returns the proportion of the data in
a normal distribution that is between xmin and xmax
• Xmin is the smallest x value in the range
 Xmin = –10 for ‘less than’, ‘no more than’
• Xmax is the largest x value in the range
 Xmax = 10 for ‘at least’ , ‘greater than’
• Mean = mean of the distribution
• Sd = standard deviation of distribution
Normalcdf(xmin,xmax)
• Returns the proportion of the data in a Standard Normal
distribution that is between xmin and xmax
• Xmin is the smallest x value in the range
 Xmin = –10 for ‘less than’, ‘no more than’
• Xmax is the largest x value in the range
 Xmax = 10 for ‘at least’ , ‘greater than’
invNorm(prob)
• Returns, in a standard normal distribution,
the cutoff point that has an area of prob to
the left.
• Prob = the proportion to the left of the
cutoff where 0≤prob≤1
• If prob < .5 then the cutoff returned will be
negative