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Normal Distribution: Calculating Probabilities/Areas (z-table) - YouTube
Channel: Joshua Emmanuel
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Welcome!
In this video, I’ll be showing how to use
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the standard normal tables to calculate the
probabilities in a normal distribution.
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A normal distribution is a symmetric, bell-shaped
distribution where the area under the normal
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curve is 1 or 100%.
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The standard normal distribution,
or what is also called the z distribution,
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is a special normal distribution with a mean
(µ) of 0 and a standard deviation (σ) of1.
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The formula for transforming a score or observation
x from any normal distribution to a standard
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normal score is z=(x-μ)/σ
The standard normal score (also known as the
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z-score or z-value) is the number of standard
deviations a score x is from the mean.
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The standard normal tables we will be using
are the “Less Than” cumulative tables.
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They usually have the left tail of the distribution
shaded, and also have positive and negative
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parts.
Let’s look at an example.
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Scores on an exam are normally distributed
with a mean of 65 and a standard deviation
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of 9. We want to find the percent of scores
satisfying a), b), and c) here.
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In a), we want the probability that x is less
than 54.
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So for x =54, the corresponding z-score is
54 minus 65 divided by 9. And that gives -1.2222
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repeating.
Since the z table is set up to handle only
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2 decimal places, we round this to -1.22.
We then go to the z-table and look up the
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area.
For z = -1.22, we go to the negative side
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of the table, look for -1.2 in the first column
and 0.02 at the top.
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The corresponding area here is 0.1112.
That is, the area to the left a z-score of
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-1.22 is 0.1112.
So on this normal curve, for z = -1.22, the
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area on the left here is 0.1112 as seen on
the table.
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Therefore, the probability that x is less
than 54 is the probability that z is less
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than -1.22 which gives 0.1112 or 11.12%.
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In b) we want the probability that x is at
least 80.
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In continuous distributions, like the normal
distribution, there is no distinction between
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“x is at least 80” and “x is greater
than 80”. We apply the same approach in
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both cases.
So for z =80,
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Z equals 80 minus 65 divided by 9. And that
gives 1.67, to 2 decimal places.
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When we look that up in the z-table by checking
1.6 under .07, we find 0.9525 which is the
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area to the left of z here.
We then subtract it from 1 to obtain the greater
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than area, since the total area under the
curve is 1.
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Therefore, the probability that x is at least
80 is the probability that z is greater than
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1.67 which equals 1 - 0.9525, giving 0.0475
or 4.75%.
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In c) we want the probability that x is between
70 and 86.
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For x =70, z equals 0.56
And for z =86, z equals 2.33.
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On the table, the area less than z = 2.33
is 0.9901.
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While the area less than z = 0.56 is 0.7123.
When finding the area between two z values
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from the cumulative Less Than tables, we simply
subtract the smaller area from the larger
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one.
So the probability that x is between 70 and
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86 is the probability that z is between .56
and 2.33. That is, 0.9901 minus 0.7123 which
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gives 0.2778 or 27.78%.
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In summary, if you’re finding a “less
than” area, using the cumulative Less Than
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table, the area in the table is the answer.
If you want a greater than area, then do 1
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minus the area from the table.
And if you want the area between two z values,
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then do bigger area (which will correspond
to the larger z value) minus the smaller area
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(which will correspond to the smaller z-value).
Note that we do not subtract z values, we
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only subtract areas.
And that concludes this video.
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Thanks for watching.
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