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2.2: Standard Normal Calculations
Standardizing: ALL normal distributions are the SAME if we measure in units of size s
about the mean µ.
How to Standardize a Distribution
If x is an observation from a distribution of mean µ and standard deviation s, the
standardized value of x is:
A standardized value is often called a z-score.
Standardizing yields the Standard Normal Curve. This is a normal distribution with a
mean of 0 and a standard deviation of 1. We denote that as: N(0,1).
Let’s look at a picture of it now on the TI-83.
Some general info:
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Observations larger than µ are positive
Observations smaller than µ are negative
A z-score is a measure of how many standard deviations away the observation is
A standard normal curve is denoted N(0, 1)
Example: The heights of young women are approximately normal with µ = 64.5 inches
and s = 2.5 inches. Find the standardized height of a young woman that is 68 inches tall
and then of someone 60 inches tall and interpret your results.
Solution:
A woman 68 inches tall is 1.4 standard deviations above the mean height of 64 inches.
A woman 60 inches tall is 1.8 standard deviations less than the mean height of 64 in.
Suppose instead of looking at just the height of one woman, we wanted to know what
proportion of ALL WOMEN were less than 68 inches. Standardizing rescues us again!
Example: Find the percentage of young women less than 68 inches tall.
Now we look at table A in the front of your book.
We find “1.40” on the second page and see the entry is .9192. This means almost 92% of
all women are shorter than 68 inches.
We can represent this graphically too.
ShadeNorm(lower bound, upper bound, mean, standard deviation)
Press “WINDOW.”
xmin = 55; xmax = 75, x-scale = 3
ymin = -.1, ymax = .5, y-scale = .1
ShadeNorm(-1e99, 68, 64.5, 2.5) and press “ENTER” (not GRAPH)
Your turn: Find the percentage of all women who are:
a. Shorter than 60 inches.
b. Taller than 66 inches.
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To find the area of the curve LESS THAN a z-score, use the table in the front of
your book.
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To find the area of the curve GREATER THAN a z-score, use 1 - Prob(z-score).
What this is all really telling you is the PROBABILITY that something will occur.
By finding the relative frequency below a z-score, you’re finding the probability that your
event happens.
Hence, why notation like P(Z < -1.4) = .0808 is used. Read aloud, this means: Probability
that Z is less than a z-score of -1.4 is .0808, or 8.08%.
Another way to say this is, The probability that an observation falls more than 1.4
standard deviations below the mean is 0.0808, or 8%.