Download Introduction to Probability and Statistics Eleventh Edition

CHAPTER 6
The Normal Probability
Distribution
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Properties of Continuous
Probability Distributions
• The area under the curve is equal to 1.
• P(a  x  b) = area under the curve between a and b.
•There is no probability attached to any single value of
x. That is, P(x = a) = 0.
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Example
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?
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The Normal Distribution
• The formula that generates the normal
probability distribution is:
1  x 
 

2  
2
1
f ( x) 
e
for   x 
 2
e  2.7183
  3.1416
 and  are the population mean and standard deviation.
• The shape and location of the normal curve changes
as the mean and standard deviation change.
MY
APPLET
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Using standard normal Table
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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Finding Probabilities for the
General Normal Random Variable
To find an area for a normal random variable x with
mean  and standard deviation , standardize or rescale
the interval in terms of z.
Find the appropriate area using Table 3.
Example: x has a normal
distribution with  = 5 and  = 2.
Find P(x > 7).
75
P ( x  7)  P ( z 
)
2
 P( z  1)  1  .8413  .1587
1
z
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Example
The weights of packages of ground beef are normally
distributed with mean 1Kg and standard deviation
0.15Kg. What is the probability that a randomly
selected package weighs between 0.7 and 0.775 Kgs?
What is the weight of a package such
that only 1% of all packages exceed
this weight?
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CHAPTER 7
Sampling Distributions
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Introduction
• Parameters are numerical descriptive measures for
populations.
– For the normal distribution, the location and shape
are described by  and .
– 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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Sampling Distributions
•Numerical
descriptive measures calculated from the
sample are called statistics.
•Statistics
vary from sample to sample and hence are
random variables.
•The
probability distributions for statistics are called
sampling distributions.
•In
repeated sampling, they tell us what values of the
statistics can occur and how often each value occurs.
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Example:
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.
X

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Central Limit Theorem
If random samples of n observations are drawn from any
population with finite  and standard deviation  , then,
when n is large, the sampling distribution of X is
approximately normal, with mean  and standard deviation
 / n . The approximation becomes more accurate as n
becomes large.

1.
2.
3.
4.
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  and variance σ2/n.
• The sum of n measurements is approximately
normal with mean n 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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How Large does n need to be?
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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The CLT in the real-world:
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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Example
Suppose X is a binomial random variable with
n = 30 and p = .4. Use the normal approximation
to find P(X  10).
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Example
A production line produces AA batteries with a
reliability rate of 95%. A sample of n = 200 batteries
is selected. Find the probability that at least 195 of the
batteries work.
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Example
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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Example
The soda bottler in the previous example claims that only
5% of the soda cans are under filled. A quality control
technician randomly samples 200 cans of soda. What is the
probability that more than 10% of the cans are under filled?
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