Chapter
37
The Normal
Probability
Distribution
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Section 7.4 Assessing Normality
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7-2
Suppose that we obtain a simple random sample from a
population whose distribution is unknown. Many of the
statistical tests that we perform on small data sets (sample
size less than 30) require that the population from which the
sample is drawn be normally distributed. Up to this point,
we have said that a random variable X is normally
distributed, or at least approximately normal, provided the
histogram of the data is symmetric and bell-shaped. This
method works well for large data sets, but the shape of a
histogram drawn from a small sample of observations does
not always accurately represent the shape of the population.
For this reason, we need additional methods for assessing
the normality of a random variable X when we are looking at
sample data.
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7-4
A normal probability plot plots observed data
versus normal scores.
A normal score is the expected Z-score of the
data value if the distribution of the random
variable is normal. The expected Z-score of an
observed value will depend upon the number of
observations in the data set.
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7-6
The idea behind finding the expected Z-score is that if the data
comes from a population that is normally distributed, we should
be able to predict the area left of each of the data values. The
value of fi represents the expected area left of the ith data value
assuming the data comes from a population that is normally
distributed. For example, f1 is the expected area left of the
smallest data value, f2 is the expected area left of the second
smallest data value, and so on.
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If sample data is taken from a population that is
normally distributed, a normal probability plot of the
actual values versus the expected Z-scores will be
approximately linear.
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We will be content in reading normal probability plots
constructed using the statistical software package,
MINITAB. In MINITAB, if the points plotted lie within
the bounds provided in the graph, then we have
reason to believe that the sample data comes from a
population that is normally distributed.
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EXAMPLE
Interpreting a Normal Probability Plot
The following data represent the time between eruptions
(in seconds) for a random sample of 15 eruptions at the
Old Faithful Geyser in California. Is there reason to
believe the time between eruptions is normally
distributed?
728
730
726
678
722
716
723
708
736
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735
708
736
735
714
719
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The random variable “time between eruptions” is
likely not normal.
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EXAMPLE
Interpreting a Normal Probability Plot
Suppose that seventeen randomly selected workers at a detergent factory
were tested for exposure to a Bacillus subtillis enzyme by measuring the ratio
of forced expiratory volume (FEV) to vital capacity (VC). NOTE: FEV is the
maximum volume of air a person can exhale in one second; VC is the
maximum volume of air that a person can exhale after taking a deep breath. Is
it reasonable to conclude that the FEV to VC (FEV/VC) ratio is normally
distributed?
Source: Shore, N.S.; Greene R.; and Kazemi, H. “Lung Dysfunction in Workers Exposed to Bacillus subtillis Enzyme,”
Environmental Research, 4 (1971), pp. 512 - 519.
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Reasonable to believe that FEV/VC is normally
distributed.
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