Download standard normal curve

Stats
The Normal Distribution
6.4-6.6
Normally Distributed Variables
• Density curves—the curve that represents the
distribution
– A density curve is always above or on the horizontal
axis (can’t have a negative frequency)
– The TOTAL area under the density curve is 1
• *The percentage of all possible observations of
the variable that lie within a specific range
equals (approximately) the corresponding area
under the density curve, expressed as a
percentage
The Normal Curve (Bell Curve)
• Lots of things are “approximately” normal
(follow the general normal curve).
• A normal curve is completely determined by
the mean and standard deviation (known as
parameters)
• Pg 297 figure 6.21 and 6.22
Standardizing
• We standardize or normalize to get an even
ground for comparison…when standardized:
 1
 0
and
• When we standardize a variable, we get the
standard normal curve (not just a normal curve)
• Standardize by finding z-scores for the variable
(a z-score is an expression of how many standard deviations
the value is away from the mean)
.
z
x

Normal Compared to Standard Normal
• Standard normal curves are normal curves that
have been shifted so the mean is at 0 and the
standard deviation is 1 (a non standard normal
curve can have any value for a mean and
standard deviation).
• We can use standard normal curves to find the
percentages of ANY normal distribution that lie
between two values.
• We can use standard normal curves to compare
different normal distributions
Area Under the Standard Normal
Curve
• Standard Normal Curve Properties (z-curve)
– Total area under is 1
– Extends indefinitely in both directions
– Symmetric about 0 (mean) has a standard
deviation of 1
– Almost all the area under the standard normal
curve lies between -3 and 3 (this would be the
three standard deviation rule)
Table C
• Standard normal curve is so important that it
z-scores
has its own table…
Corresponding area
to the LEFT of the
z-score
Examples
•
•
•
•
•
•
Pg 304 ex 6.29
Pg 304 ex 6.30
Pg 305 ex 6.31
Pg 307 ex 6.34
Pg 308 ex 6.35
Pg 309 ex 6.36
– If the area we need is not exactly in table C, then we
pick the closest area in the table and use that z-score.
If the desired area is exactly in the middle of two
areas in table C, we take the mean of the two
corresponding z-scores and use that value.
68.26—95.44—99.74 Emperical Rule
• Similar to the 3-standard deviation rule, but this
is specific for a normal distribution
• Pg 298 figure 6.24
• Procedure for finding an observed value for a
specific percentage
–
–
–
–
Sketch normal curve
Shade region of interest
Use table C to determine desired z-score(s)
Find the x-values (using z-score formula) for desired zscores
z
z
This notation is used to denote the z-score
that has an area of α to the RIGHT (this
notation is used in a later formula)
Find z0.3:
So z0.3 is wanting to know the z-score that
has an area of 0.3 to the right
Working With Normally Distributed
Variables
• General guidelines
– 1. sketch the normal curve
– 2. shade the area of interest and mark its x-values
– 3. find the corresponding z-scores
– 4. Use table C to find the desired area
• Pg 314 ex6.39