Chapter 8
Estimation
Understandable Statistics
Ninth Edition
By Brase and Brase
Prepared by Yixun Shi
Bloomsburg University of Pennsylvania
Estimating µ When σ is Known
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Point Estimate
• An estimate of a population parameter given by
a single number.
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Margin of Error
• Even if we take a very large sample size,
will differ from µ.
x
Margin of Error x  
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Confidence Levels
• A confidence level, c,
is any value between
0 and 1 that
corresponds to the
area under the
standard normal
curve between –zc
and +zc.
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Critical Values
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Common Confidence Levels
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Recall From Sampling Distributions
• If we take samples of size n from our
population, then the distribution of the sample
mean has the following characteristics:
Mean of x  x  x
Standard Deviationof x  x  x
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n
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A Probability Statement
• In words, c is the probability that the sample
mean will differ from the population mean by at
most
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Maximal Margin of Error
• Since µ is unknown, the margin of error | x - µ|
is unknown.
• Using confidence level c, we can say that
differs from µ by at most:
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x
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Confidence Intervals
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Critical Thinking
• Since x is a random variable, so are the
endpoints x  E
• After the confidence interval is numerically fixed
for a specific sample, it either does or does not
contain µ.
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Critical Thinking
• If we repeated the confidence interval process
by taking multiple random samples of equal size,
some intervals would capture µ and some would
not!
• Equation
states that the
proportion of all intervals containing µ will be c.
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Multiple Confidence Intervals
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Estimating µ When σ is Unknown
• In most cases, researchers will have to estimate
σ with s (the standard deviation of the sample).
• The sampling distribution for x will follow a
new distribution, the Student’s t distribution.
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The t Distribution
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The t Distribution
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The t Distribution
• Use Table 6 of
Appendix II to find the
critical values tc for a
confidence level c.
• The figure to the right
is a comparison of
two t distributions and
the standard normal
distribution.
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Using Table 6 to Find Critical Values
• Degrees of freedom, df, are the row headings.
• Confidence levels, c, are the column headings.
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Maximal Margin of Error
• If we are using the t distribution:
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What Distribution Should We Use?
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Estimating p in the Binomial Distribution
• We will use large-sample methods in which the
sample size, n, is fixed.
• We assume the normal curve is a good
approximation to the binomial distribution if both
np > 5 and nq = n(1-p) > 5.
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Point Estimates in the Binomial Case
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Margin of Error
• The magnitude of the difference between the
actual value of p and its estimate pˆ is the
margin of error.
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The Distribution of pˆ
• The distribution is well approximated by a
normal distribution.
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A Probability Statement
With confidence level c, as before.
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Public Opinion Polls
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Choosing Sample Sizes
• When designing statistical studies, it is good
practice to decide in advance:
– The confidence level
– The maximal margin of error
• Then, we can calculate the required minimum
sample size to meet these goals.
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Sample Size for Estimating μ
• If σ is unknown, use σ from a previous study or
conduct a pilot study to obtain s.
Always round n up to the next integer!!
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Sample Size for Estimating pˆ
If we have no preliminary estimate for p, use the following modification:
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Independent Samples
• Two samples are independent if sample data
drawn from one population is completely
unrelated to the selection of a sample from the
other population.
– Occurs when we draw two random samples
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Dependent Samples
• Two samples are dependent if each data value
in one sample can be paired with a
corresponding value in the other sample.
– Occur naturally when taking the same
measurement twice on one observation
• Example: your weight before and after
the holiday season.
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Confidence Intervals for
μ1 – μ2 when σ1, σ2 known
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Confidence Intervals for
μ1 – μ2 when σ1, σ2 known
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Confidence Intervals for
μ1 – μ2 when σ1, σ2 unknown
• If σ1, σ2 are unknown, we use the t distribution
(just like the one-sample problem).
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What if σ1 = σ2 ?
• If the sample standard deviations s1 and s2 are
sufficiently close, then it may be safe to assume
that σ1 = σ2.
– Use a pooled standard deviation.
– See Section 8.4, problem 27.
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Summarizing Intervals for
Differences in Population Means
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Estimating the Difference in Proportions
• We consider two independent binomial
distributions.
• For distribution 1 and distribution 2,
respectively, we have:
n1
p1
q1
r1
n2
p2
q2
r2
• We assume that all the following are greater
than 5:
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Estimating the Difference in Proportions
r1 r 2
Then  has the following properties :
n1 n 2
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Critical Thinking
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