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11/9/10
The Normal Probability
Distribution
• There is no probability attached to any single value of
x. That is, P(x = a) = 0.
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Copyright ©2011 Nelson Education Limited
•  The formula that generates the normal
probability distribution is:
Suppose that a random variable X is
uniformly distributed on the interval
[0,3]. That is, its density is some
constant c on [0,3], and is zero
otherwise. What is the value of c?
•  The shape and location of the normal curve changes
as the mean and standard deviation change.
MY
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The four digit probability in a particular row and column
of Table 3 gives the area under the z curve to the left that
particular value of z.
Area for z = 1.36
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APPLET
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 To find an area for a normal random variable x with
mean µ and standard deviation σ, standardize or rescale
the interval in terms of z.
 Find the appropriate area using Table 3.
1
z
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Sampling Distributions
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Copyright ©2011 Nelson Education Limited
•  Parameters are numerical descriptive measures for
populations.
–  For the normal distribution, the location and shape
are described by µ and σ.
–  For a binomial distribution consisting of n trials,
the location and shape are determined by p.
•  Often the values of parameters that specify the exact
form of a distribution are unknown.
•  You must rely on the sample to learn about these
parameters.
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A population is made up of the numbers 3,5,2,1.
You draw a sample of size n=3 without
replacement and calculate the sample average.
Find the distribution of Xbar, the sample average.
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If random samples of n observations are drawn from any
population with finite µ and standard deviation σ , then,
when n is large, the sampling distribution of X is
approximately normal, with mean µ and standard deviation
. The approximation becomes more accurate as n
becomes large.
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any population
random sampling (independence)
if start with normal, this is exact!
how large does n have to be?
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Many (most?) statistics used are sums or averages.
The CLT gives us their sampling distributions.
Overall, we have that
•  The average of n measurements is approximately
normal with mean µ and variance σ2/n.
•  The sum of n measurements is approximately
normal with mean nµ and variance nσ2.
•  The sample proportion is approximately normal with
mean p and variance p(1-p)/n.
•  The binomial is approximately normal with mean np
and variance np(1-p).
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•  If
the original distribution is normal, then any
sample size will do.
•  If
the sample distribution is approximately normal,
then even small n will work.
• If
the sample distribution is skewed, then a larger n
is needed (eg. n bigger than 30).
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1.  Check the random sampling – was it done
properly? Are your observations independent?
2.  Look at a histogram of your data to see if it is
skewed or symmetric. The more skewed the
data, the less credible the approximation. Is
your sample size big enough?
Rules of thumb:
a.  In general, use n≥30 for skewed distributions
b.  Use np, n(1-p)≥ 5 for the binomial/sample
proportion.
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Copyright ©2011 Nelson Education Limited
A bottler of soft drinks packages cans in six-packs.
Suppose that the fill per can has an approximate normal
distribution with a mean of 355 ml and a standard
deviation of 5.91 ml. What is the probability that the
total fill for a case is less than 347.79 ml?
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