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Chapter 8
Estimation
Understanding Basic Statistics
Fifth Edition
By Brase and Brase
Prepared by Jon Booze
Estimating µ When σ is Known
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8|2
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,
may 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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Critical Values
Which of the following
correctly expresses the
confidence interval shown
at right?
–2.58
0
2.58
z
a). P 0.99 z1  2.58
b). P 2.58 z2.58   0.99
c). P 0 z0.99  5.16


d). P 2.58 z2.58  0.01


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Critical Values
Which of the following
correctly expresses the
confidence interval shown
at right?
–2.58
0
2.58
z
a). P 0.99 z1  2.58
b). P 2.58 z2.58   0.99
c). P 0 z0.99  5.16


d). P 2.58 z2.58  0.01


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8|8
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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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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The 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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8 | 13
Confidence Intervals
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For a population of domesticated geese, the
standard deviation of the mass is 1.3 kg. A
sample of 45 geese has a mean mass of 5.7
kg.
Find the confidence interval for the
population mean at the 95% confidence
level.
a). 5.32 <  < 6.08
b). 0 <  < 2.97
c). 5.20 <  < 6.20
d). 5.38 <  < 6.02
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For a population of domesticated geese, the
standard deviation of the mass is 1.3 kg. A
sample of 45 geese has a mean mass of 5.7
kg.
Find the confidence interval for the
population mean at the 95% confidence
level.
a). 5.32 <  < 6.08
b). 0 <  < 2.97
c). 5.20 <  < 6.20
d). 5.38 <  < 6.02
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8 | 17
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!
• The equation
P x E    x E  c








states that the proportion of all intervals
containing µ will be c.
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Interpretation of the Confidence
Interval
At the 0.90
confidence level,
1 in 10 samples,
on average, will
fail to enclose
the true mean 
within the
confidence
interval.
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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
non-normal distribution called the Student’s t
distribution.
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The t Distribution
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The t Distribution
Find the t-value for the
following data:
x  55.2,   58.1, s  4.2, n  40
a). –27.62
b). –0.11
c). –8.95
d). –4.37
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x 
t
s
n
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The t Distribution
Find the t-value for the
following data:
x  55.2,   58.1, s  4.2, n  40
a). –27.62
b). –0.11
c). –8.95
d). –4.37
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x 
t
s
n
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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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Using Table 4 to Find Critical Values
Use Table 4 in the Appendix to find the critical value
tc for a 0.95 confidence level for a t-distribution with
sample size n = 32.
a). 2.457
b). 2.438
c). 2.042
d). 2.030
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Using Table 4 to Find Critical Values
Use Table 4 in the Appendix to find the critical value
tc for a 0.95 confidence level for a t-distribution with
sample size n = 32.
a). 2.457
b). 2.438
c). 2.042
d). 2.030
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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 p̂ is the
margin of error.
the margin of error is pˆ  p
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The Distribution of p̂
• For large samples, 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 p̂
If we have no preliminary estimate for p, use the
following modification:
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8 | 42
Sample Size for Estimating p̂
How many students should be surveyed to
determine the proportion of students who
prefer vanilla ice cream to chocolate,
accurate to 0.1 at a 90% confidence level?
a). 100
b). 69
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c). 52
d). 5
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Sample Size for Estimating p̂
How many students should be surveyed to
determine the proportion of students who
prefer vanilla ice cream to chocolate,
accurate to 0.1 at a 90% confidence level?
a). 100
b). 69
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c). 52
d). 5
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