Chapter 8 Estimation
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Estimating µ when σ is know
Estimating µ when σ is unknow
Estimating p in the binomial distribution
Estimating µ1- µ2 and p1-p2
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8.1 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, x
will differ from µ.
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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Example
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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 Deviation
of 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 x
differs from µ by at most:
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Confidence Intervals
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Example
Julia enjoys jogging. She has been jogging over a
period of several years, during which time her
physical condition has remained constantly good.
Usually, she jogs 2 miles per day. The standard
deviation of her times is 1.80 minutes. During the
past year, Julia has recorded her times to run 2
miles. She has a random sample of 90 of these
times. For these 90 times, the mean was 15.60
minutes. Let µ be the mean jogging time for the
entire distribution of Julia’s 2-mile running times
(taken over the past year). Find a 0.95 confidence
interval for µ .
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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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Sample Size for Estimating the
Mean µ
x
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Sample Size for Estimating the
Mean µ
If n is not a whole number, increase n to the next
higher whole number. Note that n is the minimal
sample size for a specified confidence level and
maximal error of estimate E.
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Example
A wildlife study is designed to find the mean weight
of salmon caught by an Alaskan fishing company.
A preliminary study of a random sample of 50
salmon showed s=2.15 pounds. How large a
sample should be taken to be 99% confi dent that
the sample mean is within 0.20 pound of the true
mean weight µ?
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8.2 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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Example
Find the critical value t c for a 0.99 confidence
level for a t distribution with sample size n=5.
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Maximal Margin of Error
• If we are using the t distribution:
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Example
Suppose an archaeologist discovers only seven fossil
skeletons from a previously unknown species of
miniature horse. Reconstructions of the skeletons of
these seven miniature horses show the shoulder
heights (in centimeters) to be
45.3 47.1 44.2 46.8 46.5 45.5 47.6
For these sample data, the mean is 46.14 and the
sample standard deviation is s=1.19. Let µ
be the mean shoulder height (in centimeters) for this
entire species of miniature horse, and assume that
the population of shoulder heights is approximately
normal. Find a 99% confidence interval for µ, the
mean shoulder height of the entire population of
such horses.
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What Distribution Should We Use?
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8.3 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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8.4 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
Then
r1
n1

r2
has the following
properties
:
n2
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Critical Thinking
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