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Normal Distribution
Section 2.2
Objectives



Introduce the Normal Distribution
Properties of the Standard Normal
Distribution
Use Normal Distribution in an inferential
fashion
Theoretical Distribution

Empirical distributions


based on data
Theoretical distribution

based on mathematics

derived from model or estimated from data
Normal Distribution
Why are normal distributions so important?
 Many variables are commonly assumed to be
normally distributed in the population


Height, weight, IQ scores, ACT scores, etc.
If a variable is approximately normally
distributed we can make inferences about
values of that variable
Normal Distribution


Since we know the shape of the curve, we can
calculate the area under the curve
The percentage of that area can be used to determine
the probability that a given value could be pulled
from a given distribution

The area under the curve tells us about the probability- in
other words we can obtain a p-value for our result (data)
by treating it as a normally distributed data set.
Key Areas under the Curve

For normal
distributions
+ 1 SD ~ 68%
+ 2 SD ~ 95%
+ 3 SD ~ 99.7%
Standard Normal Distributions
Standard Normal Distribution – N(, )




We agree to use the
standard normal
distribution
Roughly symmetric
=0
=1
Recall: Z-score
If we know the population mean and
population standard deviation, for any value
of X we can compute a z-score by subtracting
the population mean and dividing the result by
the population standard deviation
z
X 

Proportions




Total area under the curve
is 1
The area in red is equal to
p(z > 1)
The area in blue is equal to
p(-1< z <0)
Since the properties of the
normal distribution are
known, areas can be looked
up on tables or found with
a calculator.
Suppose Z has standard normal distribution
Find 0 < Z < 1.23
Find -1.57 < Z < 0
Find Z > 0.78
Z is standard normal
Calculate -1.2 < Z < 0.78
Work time...

What is the area for scores less than z = -1.5?

What is the area between z =1 and 1.5?

What z-score cuts off the highest 30% of the
distribution?

What two z-scores enclose the middle 50% of the
distribution?