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Chapter 7 Introduction to Sampling
Distributions
• Sampling Distributions
• The Central Limit Theorem
• Sampling Distributions for Proportions
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7.1 Sampling Distributions
• Statistic
A statistic is a numerical descriptive measure of a
sample.
• Parameter
A parameter is a numerical descriptive measure of
a population.
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Terms, Statistics & Parameters
• Terms: Population, Sample, Parameter,
Statistics
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Why Sample?
• At times, we’d like to know something about the
population, but because our time, resources,
and efforts are limited, we can take a sample to
learn about the population. In such cases, we
will use a statistic to make inferences about a
corresponding population parameter. The
followings are the principal types of inferences.
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Types of Inference
1) Estimation: We estimate the value of a
population parameter.
2) Testing: We formulate a decision about a
population parameter.
3) Regression: We make predictions about the
value of a statistical variable.
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Sampling Distributions
• To evaluate the reliability of our inference, we
need to know about the probability distribution
of the statistic we are using.
• Typically, we are interested in the sampling
distributions for sample means and sample
proportions.
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Examples
• Example 1/p295.
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7.2 The Central Limit Theorem
• If x is a random variable with a normal
distribution, mean = µ, and standard deviation =
σ, then the following holds for any sample size:
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Example
Suppose a team of biologists has been studying
the Pinedale children’s fishing pond. Let x
represent the length of a single trout taken at
random from the pond.This group of biologists has
determined that x has a normal distribution with
mean m10.2 inches and standard deviation s1.4
inches.
(a) What is the probability that a single trout taken
at random from the pond is between 8 and 12
inches long?
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Example
b) What is the probability that the mean length of
five trout taken at random is between 8 and 12
inches?
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The Standard Error
• The standard error is just another name for the
standard deviation of the sampling distribution.
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The Central Limit Theorem
(Any Distribution)
• If a random variable has any distribution with
mean = µ and standard deviation = σ, the
sampling distribution of x will approach a
normal distribution with mean = µ and standard
deviation =  n as n increases without limit.
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Sample Size Considerations
• For the Central Limit Theorem (CLT) to be
applicable:
– If the x distribution is symmetric or
reasonably symmetric, n ≥ 30 should suffice.
– If the x distribution is highly skewed or
unusual, even larger sample sizes will be
required.
– If possible, make a graph to visualize how
the sampling distribution is behaving.
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Critical Thinking
• Bias – A sample statistic is unbiased if the
mean of its sampling distribution equals the
value of the parameter being estimated.
• Variability – The spread of the sampling
distribution indicates the variability of the
statistic.
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7.3 Sampling Distributions for Proportions
r
pˆ 
n
• If np > 5 and nq > 5, then p̂ can be
approximated by a normal variable with mean
and standard deviation  pˆ  p
and
pq
 pˆ 
n
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Continuity Corrections
• Since p̂ is discrete, but x is continuous, we
have to make a continuity correction.
• For small n, the correction is meaningful.
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Examples
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Examples
The annual crime rate in the Capital Hill
neighborhood of Denver is 111 victims per 1000
residents. This means that 111 out of 1000
residents have been the victim of at least one
crime (Source:Neighborhood Facts, Piton
Foundation). These crimes range from relatively
minor crimes (stolen hubcaps or purse snatching)
to major crimes (murder). The Arms is an
apartment building in this neighborhood that has
50 year-round residents.Suppose we view each of
the n=50 residents as a binomial trial. The random
variable r (which takes on values 0, 1, 2, . . . , 50)
represents the number of victims of at least one
crime in the next year.
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Example
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Control Charts for Proportions
• Used to examine an attribute or quality of an
observation (rather than a measurement).
• We select a fixed sample size, n, at fixed time
intervals, and determine the sample proportions
at each interval.
• We then use the normal approximation of the
sample proportion to determine the control
limits.
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Procedure
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(c) Signal III: at least two out of three consecutive
points beyond a control limit (on the same side).
If no out-of-control signals occur, we say that the
process is “in control,”
while keeping a watchful eye on what occurs next.
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Example
Anatomy and Physiology is taught each semester.
The course is required for several popular healthscience majors, so it always fills up to its maximum
of 60 students. The dean of the college asked the
biology department to make a control chart for the
proportion of A’s given in the course each
semester for the past 14 semesters. Using
information from the registrar’s office, the following
data were obtained. Make a control chart and
interpret the result.
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Example
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Example
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P-Chart Example
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Example
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Example
(e) Conclusion: The biology department can tell
the dean that the proportion of A’s given in
Anatomy and Physiology is in statistical control,
with the exception of one unusually good class two
semesters ago.
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