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The Statistical Imagination
• Chapter 8. Parameter Estimation
Using Confidence Intervals
© 2008 McGraw-Hill Higher Education
Confidence Intervals (CI)
• A range of possible values of a
parameter expressed with a
specific degree of confidence
• Confidence interval = point
estimate ± error term
© 2008 McGraw-Hill Higher Education
With a Confidence
Interval (CI):
• We take a point estimate and use knowledge
about sampling distributions to project an interval
of error around it
• A CI provides an interval estimate of an unknown
population parameter and precisely expresses the
confidence we have that the parameter falls within
that interval
• Answers the question: What is the value of a
population parameter, give or take a little known
sampling error?
© 2008 McGraw-Hill Higher Education
The Level of Confidence
• The level of confidence is a
calculated degree of confidence that
a statistical procedure conducted with
sample data will produce a correct
result for the sampled population
© 2008 McGraw-Hill Higher Education
The Level of Significance (α)
• The level of significance is the difference
between the stated level of confidence and
“perfect confidence” of 100%
• This is also called the level of expected
error
• The Greek letter alpha (α) is used to
symbolize the level of significance
© 2008 McGraw-Hill Higher Education
Confidence and Significance
• The level of confidence and the level of
significance are inversely related – as one
increases, the other decreases
• The level of confidence plus the level of
significance sum to 100%. E.g., a level of
confidence of 95% has a level of
significance of 5%, or a proportion of .05
© 2008 McGraw-Hill Higher Education
The Critical Z-score
• We choose a desired level of confidence
by selecting a critical Z-score from the
normal distribution table
• This critical score fits the normal curve
and isolates the area of the level of
confidence and significance
• Use the symbol, Zα, for critical scores
© 2008 McGraw-Hill Higher Education
Commonly Used
Critical Z-scores
• For a 95% CI of the mean, when n >
121, the critical Z-score = 1.96 SE
• For a 99% CI of the mean, when n >
121, the critical Z-score = 2.58 SE
• For a CI of the mean, when n < 121,
the critical value is found in a tdistribution table with df = n – 1 (See
Chapter 10.)
© 2008 McGraw-Hill Higher Education
Steps for Computing
Confidence Intervals
• Step 1. State the research question; draw a
conceptual diagram depicting givens (e.g.,
Figure 8-1 in the text);
• Step 2. Compute the standard error and the
error term
• Step 3. Compute the LCL and UCL of the CI
• Step 4. Provide an interpretation in everyday
language
• Step 5. Provide a statistical interpretation
© 2008 McGraw-Hill Higher Education
When to Calculate a CI
of a Population Mean
• The research question calls for
estimating the population parameter μX
• The variable of interest (X) is of
interval/ratio level
• There is a single representative
sample from one population
© 2008 McGraw-Hill Higher Education
The Error Term
• The error term of the CI is calculated
by multiplying a standard error by a
critical Z-score
• For a CI of the mean, the standard
error is the standard deviation divided
by the square root of n
© 2008 McGraw-Hill Higher Education
Upper and Lower
Confidence Limits
• The upper confidence limit (UCL)
provides an estimate of the highest
value we think the parameter could
have
• The lower confidence limit (LCL)
provides an estimate of the lowest
value we think the parameter could
have
© 2008 McGraw-Hill Higher Education
Calculating the
Confidence Limits
• UCL = sample mean + the error term
• LCL = sample mean – the error term
© 2008 McGraw-Hill Higher Education
Interpretation in
Everyday Language
• Without technical language, this is a
statement of the findings for a public
audience
• We state that we are confident to a
certain degree (e.g., 95%) that the
population parameter falls between
the limits of our confidence interval
© 2008 McGraw-Hill Higher Education
The Statistical Interpretation
• The statistical interpretation illustrates the
notion of "confidence in the procedure"
used to calculate the confidence interval
• E.g., for the 95% level of confidence we
state: If the same sampling and statistical
procedures are conducted 100 times, 95
times the true population parameter will be
encompassed in the computed intervals
and 5 times it will not. Thus, I have 95%
confidence that this single CI I computed
includes the true parameter
© 2008 McGraw-Hill Higher Education
Some Things to Note
About a CI of the Mean
• Typically, the sample standard deviation is
used to estimate the standard error (SE)
• The error term = SE times Zα . A large
error term results when either SE or Zα is
large
• The interval reported is an estimate of the
population mean, not an estimate of the
range of X-scores
© 2008 McGraw-Hill Higher Education
Level of Confidence and
Degree of Precision
• The greater the stated level of confidence,
the less precise the confidence interval
• The larger the sample size, the more
precise the confidence interval
• To obtain a high degree of precision and a
high level of confidence a researcher must
use a sufficiently large sample
© 2008 McGraw-Hill Higher Education
Confidence Interval of a
Population Proportion
• With a nominal/ordinal variable, a
confidence interval provides an
estimate within a range of error of the
proportion of a population that falls in
the “success” category of the variable
© 2008 McGraw-Hill Higher Education
When to Calculate a CI of a
Population Proportion
• We are to provide an interval estimate of
the value of a population parameter, Pµ ,
where P = p [of the success category] of a
nominal/ordinal variable
• There is a single representative sample
from one population
• The sample size is sufficiently large that
(psmaller) (n) > 5, resulting in a sampling
distribution that is approximately normal
© 2008 McGraw-Hill Higher Education
Choosing a Sample Size
• To obtain a high degree of precision and a
high level of confidence a researcher must
use a sufficiently large sample
• Sample size can be chosen to fit a desired
level of confidence and range of error
• The formula for choosing n involves solving
for n in the error term of the confidence
interval equation
© 2008 McGraw-Hill Higher Education
Statistical Follies
• Scrutinize reports of survey and poll
results. Even a major news network
may misreport results
• Often confusion centers around the
error term
• It is plus and minus the error term
© 2008 McGraw-Hill Higher Education