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MATH& 146
Lesson 19
Section 2.6
Assessing Normality
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Example 1
Cumulative SAT scores are approximated well by a
normal model, N(μ = 1500, σ = 300).
Shannon is a randomly selected SAT taker, and
nothing is known about Shannon's aptitude. What
is the probability Shannon scores at least 1630 on
her SATs?
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Tip
Always draw a picture first, and find the
probability.
For any normal probability situation, always always
always draw and label the normal curve and shade
the area of interest first. The picture will provide
an estimate of the probability.
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Example 2
Cumulative SAT scores are approximated well by a
normal model, N(μ = 1500, σ = 300).
If the probability of Shannon scoring at least 1630
is 0.3324, then what is the probability she scores
less than 1630?
Draw the normal curve representing this question,
shading the lower region instead of the upper one.
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Example 3
Cumulative SAT scores are approximated well by a
normal model, N(μ = 1500, σ = 300).
a) Edward earned a 1400 on his SAT. What is his
percentile?
b) Use the results of part (a) to compute the
proportion of SAT takers who did better than
Edward.
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Example 4
Cumulative SAT scores are approximated well by a
normal model, N(μ = 1500, σ = 300). Stuart
earned an SAT score of 2100.
a) What is his percentile?
b) What percent of SAT takers did better than
Stuart?
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Example 5
Based on a sample of 100 men, the heights of
male adults between the ages 20 and 32 in the US
is nearly normal with mean 70.0" and standard
deviation 3.3".
Mike is 5'7" and Jim is 6'4". What are Mike's and
Jim's height percentiles?
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Example 6
Based on a sample of 100 men, the heights of
male adults between the ages 20 and 32 in the US
is nearly normal with mean 70.0" and standard
deviation 3.3".
a) Erik's height is at the 40th percentile. How tall is
he?
b) What is the adult male height at the 82nd
percentile?
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Example 7
Based on a sample of 100 men, the heights of
male adults between the ages 20 and 32 in the US
is nearly normal with mean 70.0" and standard
deviation 3.3".
What is the probability that a random adult male is
between 5'9" and 6'2"?
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Assessing Normality
Many processes can be well approximated by the
normal distribution, such as SAT scores and the
heights of US adult males.
While using a normal model can be extremely
convenient and helpful, it is important to remember
normality is always an approximation.
Testing the appropriateness of the normal
assumption is a key step in many data analyses.
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Assessing Normality
There are two visual methods for checking the
assumption of normality, which can be
implemented and interpreted quickly. The first is a
simple histogram with the best fitting normal curve
overlaid on the plot, as shown in the left panel
below.
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Assessing Normality
The sample mean and standard deviation are used
as the parameters of the best fitting normal curve.
The closer this curve fits the histogram, the more
reasonable the normal model assumption.
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Assessing Normality
Another common method for assessing normality
is by examining a normal probability plot, shown
in the right panel below. The closer the points are
to a perfect straight line, the more confident we
can be that the data follow the normal model.
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Assessing Normality
Nearly Normal data have a histogram and a
normal probability plot that look somewhat like this
example:
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Assessing Normality
Data that is not normal will show a noticeable bend
in their normal probability plot.
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Assessing Normality
Reading a normal probability plot takes a little
practice. Even samples from true Normal
distributions do not produce plots that exactly
follow the straight line, especially for smaller
samples.
Indeed, samples from true Normal distributions
typically have points that meander above and
below the line, though still generally following the
line.
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Assessing Normality
Three data sets of 40, 100, and 400 samples were
simulated from a normal distribution, and graphed
below. These will provide a benchmark for what to
look for in plots of real data.
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Example 8
Describe the normality of w.
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Example 9
Describe the normality of x.
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Example 10
Describe the normality of y.
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Example 11
Describe the normality of z.
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Example 12
Are NBA player heights normally distributed?
Consider all 435 NBA players from the 2008-9
season presented below.
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Example 13
Can we approximate poker winnings by a normal
distribution? We consider the poker winnings of
an individual over 50 days.
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Example 14
The figure below shows normal probability plots for
two distributions that are skewed. One distribution is
skewed to the low end (left skewed) and the other is
skewed to the high end (right skewed). Which is
which?
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Example 14 Solution
Examine where the points fall along the vertical axis.
In the first plot, most points fall near the low end with
fewer observations scattered along the high end; this
describes a distribution that is skewed to the high end.
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Example 14 Solution
The second plot shows the opposite features, and this
distribution is skewed to the low end.
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Example 15
Is it normal for someone to shop at Safeway?
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