Download Normal probability plot

AP Statistics: Section 2.2 C
Example 1: Determine if each of the
following is likely to have a Normal
distribution (N) or a non-normal
distribution (nn).
N gas mileage of 2006 Corvettes
_____
_____
nn prices of homes in Westlake
_____
nn gross sales of business firms
_____
N weights of 9-oz bags of potato chips
While experience can suggest whether or
not a Normal distribution is plausible in a
certain case, it is very risky (especially on
the AP test) to assume that a distribution is
Normal without actually inspecting the
data. Chapters 10 -15 of our text deal with
inference, and an important condition that
will need to be met is that the data comes
from a population that is approximately
Normal.
Our goal in this section is to
explore methods for determining
if a distribution is Normal or, at
least, approximately Normal.
Method 1: Construct a histogram,
stemplot or dotplot.
What are we looking for?
bell - shaped and symmetrical
Non-Normal features to look for
include ________,
pronounced
outliers
skewness
gaps and
_________,
or ______
clusters
________.
To be truly Normal, such a bellshaped plot should conform to the
68-95-99.7 rule. Mark the points
on the horizontal axis and
determine the percentage of
counts in each interval.
1s from x : 649
947
2s from x : 903
3s from x : 945
 68.5%
947
947
 95.4%
 99.8%
YES
NOTE: In some text books, an outlier is defined as any
value more than two standard deviations away from
the mean. Thus, the 68-95-99.7 rule would suggest
that a Normal distribution might have outliers. BUT, if
we are examining the distribution of a sample and
trying to determine if the corresponding population is
Normally distributed, any outliers in the sample
would indicate that the population is probably not
Normally distributed. Why? If only 5% of the original
population could be outliers, it is highly unlikely that
our sample would actually contain any outliers.
Method 2: Examine the 5-number
summary. What are we looking for.
Most importantly, we are looking for the boxplot
to be roughly symmetrical. Then we would like to
see the box tightly grouped around the center while
the whiskers are slightly longer.
Note: What should be true about
the mean and the median in a
Normal distribution?
Roughly the same.
Method 3: Construct a Normal
probability plot. Statistical packages,
as well as TI calculators, can construct
a Normal probability plot. Before we
look at constructing a Normal
probability plot, let’s look at some
basic ideas behind constructing a
Normal probability plot.
1. Arrange the observed data values
from _________
smallest to _________.
largest
Record what percentile of the data
each value represents. For example,
the smallest observation in a set of 20
would be at the _____
5th percentile, the
second smallest would be at the _____
10th
percentile, etc.
2. Determine the z-scores for each
of the observations. For example,
z = _______
1.645 is the 5% point of the
standard Normal distribution and
z = _______
 1.28 is the 10% point.
3. Plot each data point x against
the corresponding z. If the data
distribution is close to Normal, the
plotted points will be
__________________.
approximately linear
Systematic deviations from a
straight line indicate a non-Normal
distribution.
In a right-skewed distribution, the
largest observations fall distinctly
_______
above a line drawn through the
main body of points. In a left-skewed
distribution, the smallest observations
fall distinctly ________
below a line drawn
through the main body of points.
Outliers appear as points that are far
away from the overall pattern.
Construct a Normal probability plot on your
calculator as follows:
1. Enter the data in L1. (STATS, EDIT).
2. Go to STAT PLOT (2nd Y=). ENTER. Highlight ON.
First draw a histogram to see that the data
appears to be Normally distributed. Key ZOOM 9
to get an appropriate window.
3. Go back to STAT PLOT. Change “Type” to the 6th
graph choice. For “Data List”, put in L1 and
for “Data Axis”, highlight Y.
4. Push GRAPH and key ZOOM 9 to get an
appropriate window.
The data is
approximately
Normally
distributed
The data is
right - skewed