Download 6.2 lecture notes

SECTION 6.2
THE STANDARD NORMAL
DISTRIBUTION
NORMAL PROBABILITY
DISTRIBUTION
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The mean, median and mode are the same.
The distribution is bell-shaped and symmetrical
around the mean.
The total area under the curve is equal to 1.
The left side and the right side of the normal
probability distribution extend indefinitely, never
quite touching the horizontal axis.
NORMAL DISTRIBUTION
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If a continuous random variable has a
distribution with a graph that is symmetric and
bell-shaped and it can be described by the
equation we say that is has a normal
distribution.
y=
 1 x  μ 




σ
2 

e
σ
2π
2
STANDARD NORMAL
DISTRIBUTION
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Is a normal distribution with the parameters of
µ = 0 and σ = 1. The total area under its
density curve is equal to 1.
FINDING PROBABILITIES
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This section will have you find the probability /
area under the curve to the z – score.
There are 5 unique style questions

P(a < z < b)
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P(a < z )

P(z < b)
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Zα
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P%
P(a < z < b)
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The probability that the z score is between a
and b.
P(a < z)
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The probability that the z score is greater than
a.
P(z < b)
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The probability that the z score is less than b.
b
Zα
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The z-score with an area of α to its right.
P%
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The z-score with an area from 0 to the given %.
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Change % to decimal, % = α .
Calculator – Inequality
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How to find the probability with a TI
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Go to folder DISTR (2nd Vars)
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Press 2 (normalcdf) – gives percentage of area
under a standard normal distribution.
Normalcdf(lower bound, upper bound) OR
Normalcdf( lower bound, upper bound, mean ,
standard deviation)
TABLE - Inequality
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Given the z score table the graph at the top of
the page shows you the probability / area.
Understand different z scores shade different
sides, so be careful.
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The probabilities / areas are on the inside.
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The z – score is broken up into two parts
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The whole and the tenth number are on the side
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The hundredth number is on the top
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Match the two together to find the prob / area
EXAMPLE
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Find the probability of a normal standard
deviation between -0.62 and 1.78
EXAMPLE
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Find the probability of a normal standard
deviation greater than -1.04
EXAMPLE
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Find the probability of a normal standard
deviation less than -1.04
Calculator - Zα
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Subtract : 1 – α
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Distr (2nd vars)
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InvNorm(
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Type in 1 – α
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enter
Table - Zα
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Look at the shaded region of the z table
This α connects with a shaded region to the
right, if you need the shaded region to the left
then subtract from 1, 1 – α
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Find the value inside the z score table
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Look up and to the left and find the z value
EXAMPLE
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Evaluate z0.025
Calculator - P%
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Distr button (2nd vars)
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Press number 3 invNorm
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InvNorm (area → decimal no %)
Table - P%
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Look at the middle section of the z score table
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Convert & to decimal
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Find the probability

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If exact probability is not there adjust accordingly
Move up and left for the two part and combine
them.
EXAMPLE
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Evaluate P90
EXAMPLE
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Evaluate z0.025