Sections 6.2-6.3 - Faculty

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SECTIONS 6-2 AND 6-3
THE STANDARD NORMAL
DISTRIBUTION AND
APPLICATIONS OF NORMAL
DISTRIBUTIONS
Density Curve
A density curve is the graph of a
continuous probability distribution. It
must satisfy the following properties:
1.
The total area under the curve must
equal 1.
2.
Every point on the curve must have
a vertical height that is 0 or greater.
(That is, the curve cannot fall below
the x-axis.)
Because the total area under the
density curve is equal to 1, there is a
correspondence between area and
probability.
Standard Normal Distribution
The standard normal distribution is a
normal probability distribution with ߤ = 0
and ߪ = 1. The total area under its density
curve is equal to 1.
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Notation
P( z > a)
denotes the probability that the z score is
greater than a.
P( z < a)
denotes the probability that the z score is
less than a.
P ( a < z < b)
denotes the probability that the z score is
between a and b.
Using JMP or Table A-2
JMP: Two different apps for z-scores and areas
z-table: Finds the area under the curve given a zscore
z-score: Finds the z-score associated with an area
under the curve
Table A-1: When working with a graph, avoid
confusion between z-scores and areas.
z Score: Distance along horizontal scale of the
standard normal distribution; refer to the leftmost
column and top row of Table A-2.
Area: Region under the curve; refer to the values
in the body of Table A-2.
The part of the z-score denoting hundredths is
found across the top.
Converting to a Standard
Normal Distribution
‫=ݖ‬
‫ݔ‬−ߤ
‫ ݔ‬− ‫̅ݔ‬
‫ݎ݋‬
ߪ
‫ݏ‬
Formula 6-2
All normal distributions can be converted
to standard normal distributions
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Math Ranking
It’s admissions season and you are an admissions counselor at Clayton
State University. You have the application of Joey Linders in front of you.
Joey hopes to obtain a Nursing degree. Your job is to determine whether
Joey should be admitted to the nursing program.
The SAT follows approximately a normal distribution with the following
means and standard deviations:.
SAT 2013
Mathematics
Critical Reading
ഥ
࢞
514
496
࢙
115
118
Mathematics
Reading
Joey (SAT)
410
450
How do Joey’s SAT scores compare to the
rest of the country?
Finding a z Score When Given a
Probability Using Table A-2
1. Draw a bell-shaped curve and identify
the region under the curve that
corresponds to the given probability.
2.Using the cumulative area from the left,
locate the closest probability in the body
of Table A-2 and identify the
corresponding z score.
Find what z score separates the 95
percentile.
Finding z Scores When Given
Probabilities
5% or 0.05
(z score will be positive)
Finding the 95th Percentile
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Finding z Scores When Given
Probabilities
5% or 0.05
(z score will be positive)
1.645
Finding the 95th Percentile
Nursing Program
Joey is ready to apply to the BSN
program.
Complete the questions about
Joey’s performance relative to his
peers.
Helpful Hints
Data Values
(x)
Z-Scores
(z)
Probabilities
(P)
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