Transcript 2.2: Normal Distributions

2.2: Normal Distributions
Note on Uniform
Distributions
3 Reasons why we like Normal Distributions
• Good descriptions of real data (ex: SATs,
psychological tests, characteristics of
populations…)
• Good approximations to results of many kinds of
chance outcomes.
• Many inference procedures work well for
“roughly symmetrical” distributions.
• Many data sets tend to be mound-shaped
(characteristics of biological populations)
• TI83: student heights, L1, graph
Normal Distributions
• Described by giving its
mean
and std.
deviation
•
controls the spread of a
normal curve. Figure
shows curve w/different
values of .
• Changing
w/o
changing
moves the
curve along the horizontal
axis w/o changing spread.
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Locating the standard deviation by eyeballing the curve:
“Inflection Points”
Common Properties of Normal Curves
• They all have inflection points (where change of curvature takes
place).
• E. rule only provides an approximate value for the proportion of
observations that fall within 1, 2, or 3 std. devs of the mean.
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Example #1
• Suppose that taxicabs in NYC are driven
an average of 75,000 miles per year with a
standard deviation of 12,000 miles. What
information does the empirical rule tell us?
2 Normal curves
What do you notice
about their means?
What do you notice
about their standard
deviations?
Finding Areas to the Left
Find the proportion of
observations from the
standard normal
distribution that are
less than 2.22.
That is:
Find P (z < 2.22)
Finding Areas to the Right
• Find the proportion of
observations from the
standard normal
distribution that are
greater than -2.15.
• That is: find
P (z > -2.15)
Table A Practice
Use Table A to find the proportion of observations
from a standard Normal distribution that falls in
each of the following regions. In each case,
sketch a standard Normal curve and shade the
area representing the region.
1) Z is less than or equal to -2.25
2) Z is greater than or equal to -2.25
3) Z > 1.77
4) -2.25 < z < 1.77
Example
• The mean of women is 64.5 inches, and the
standard deviation is 2.5 inches. What
proportion of all young women are less
than 68 inches tall?
Example
• The level of cholesterol in the blood is important
because high cholesterol levels may increase the
risk of heart disease. The distribution of blood
cholesterol levels in a large population of people
of the same age and sex is roughly normal. For
14 year old boys, the mean is 170 mg/dl and the
standard deviation is 30 mg/dl 2. Levels above 240
mg/dl may require medical attention. What
percent of 14-year-old boys have more than 240
mg/dl of cholesterol?
What percent of 14 year old boys have between 170
and 240 mg/dl?
Finding a value given a
proportion
• Use Table A backwards!
1) Find the given proportion in the body of
the table
2) Read the corresponding z-score
3) Unstandardize to get the observed (x)
value. Voila!
Example
• Scores on the SAT verbal test in recent
years follow approximately the N(505,110)
distribution. How high must a student
score in order to place in the top 10% of all
students taking the SAT?
Special Note….
• X is greater than is the same as x is greater than
or equal to.
• That is, there is no area under the curve where x
= 240. There may be a boy with an exact
cholesterol level of 240, but there is no area
under the curve at an exact point.
• The normal distribution is therefore an
approximation – not a description of every detail
in the exact data.
Normal Probability Plot
• TI83
• Don’t overreact to minor wiggles in the
plot
• Normality cannot be assumed if there is
skewness or outliers (don’t use Normal
distribution if these things occur)!