Download Ch. 6

Chapter 6
Estimation Procedures
Copyright © 2012 by Nelson Education Limited.
6-1
In this presentation
you will learn about:
• The logic of estimation and key terms
• How to construct and interpret interval
estimates for:
– Sample means
– Sample proportions
• Width of confidence intervals
Copyright © 2012 by Nelson Education Limited.
6-2
Basic Logic of
Estimation
• In estimation procedures, statistics
calculated from random samples are
used to estimate the value of
population parameters.
Copyright © 2012 by Nelson Education Limited.
6-3
Basic Logic of
Estimation: An Example
–If we know 71% of a random sample
drawn from a city perceive lawn pesticides
to be a source of environmental pollution,
we can estimate the percentage of all city
residents who perceive these products to
be a source of environmental pollution.
Copyright © 2012 by Nelson Education Limited.
6-4
Basic Logic of
Estimation (continued)
• Information from
samples is used to
estimate
information about
the population.
POPULATION
SAMPLE
PARAMETER
• Statistics are used
to estimate
parameters.
STATISTIC
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6-5
Basic Logic of
Estimation (continued)
• Sampling
Distribution is the link
between sample and
population.
– The value of the
parameters are
unknown but
characteristics of the
sampling distribution
are defined by
theorems (see Ch. 5).
POPULATION
SAMPLING DISTRIBUTION
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SAMPLE
6-6
Two Estimation
Procedures
1. A point estimate is a sample statistic
used to estimate a population value.
Example: A newspaper story reports that
15% of a sample of randomly selected
Canadians did not have a regular medical
doctor.
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6-7
Two Estimation
Procedures (continued)
2. Confidence intervals consist of a range
of values.
Example: Between 12% and 18% of
Canadians did not have a regular medical
doctor.
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6-8
Bias and Efficiency
•
Criteria for choosing estimators:
–
Bias: An estimator is unbiased if the mean of
its sampling distribution is equal to the
population value of interest. Sample means
and proportions are unbiased estimators
(Figure 6.1).
–
Efficiency: Extent to which the sampling
distribution is clustered around its mean.
Larger samples have greater clustering
around mean and thus more efficiency
(Figures 6.2 and 6.3).
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6-9
Bias and Efficiency
(continued)
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Bias and Efficiency
(continued)
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Constructing Confidence
Intervals for Means (for Large Samples)*
• First, set the alpha, α (probability that the
interval will be wrong).
– Example: Setting alpha equal to 0.05, a 95%
confidence level, means the researcher is willing to
be wrong 5% of the time.
• Second, find the Z score associated with alpha.
– Example; If alpha is equal to 0.05, we would place
half (0.025) of this probability in the lower tail and
half (0.025) in the upper tail of the distribution.
* The procedures for constructing confidence intervals for small samples (n<100) are
not covered in this text.
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6-12
Constructing Confidence
Intervals for Means (for Large Samples)
(continued)
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6-13
Constructing Confidence Intervals
for Means (for Large Samples)
(continued)
* In Formula 6.2 σ is replaced by s. Further, n is replaced by n-1 to correct
for the fact that s is a biased estimator of σ.
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6-14
Constructing Confidence
Intervals for Means: An Example
• A random sample of 178 Canadian
households watches TV an average of 6
hours per day, with a standard deviation of
3 (s=3).
• With alpha set to .05, the confidence
interval is:
– c.i. = 6.0 ±1.96(3/√178-1)
– c.i. = 6.0 ±1.96(3/13.30)
– c.i. = 6.0 ±1.96(.23)
– c.i. = 6.0 Copyright
± .44© 2012 by Nelson Education Limited.
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Constructing Confidence
Intervals for Means: An Example
(continued)
• We can estimate that households in Canada
average 6.0 ± .44 hours of TV watching each
day.
• Another way to state the interval:
– 5.56≤μ≤6.44
• We estimate that the population mean is greater than or
equal to 5.56 and less than or equal to 6.44.
Copyright © 2012 by Nelson Education Limited.
6-16
Constructing Confidence
Intervals for Means: An Example
(continued)
• This interval has a .05 (5%) chance of being
wrong.
– Even if the statistic is as much as ±1.96 standard
deviations from the mean of the sampling
distribution the confidence interval will still include
the value of μ.
– Only rarely (5 times out of 100) will the interval not
include μ.
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6-17
Constructing Confidence Intervals
for Proportions (for Large Samples)*
• Procedures:
1. Set alpha.
2. Find the associated Z score.
3. Substitute values into the formula for constructing
confidence intervals for sample proportions:
*The procedures for constructing confidence intervals for small samples
(n<100) are not covered in this text.
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6-18
Constructing Confidence Intervals
for Proportions (for Large Samples) (continued)
where Ps = sample proportion
Z = Z score as determined by the alpha level
Pu = population proportion (Pu is typically setting at .5)
= standard deviation of the sampling distribution
of sample proportions
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Constructing Confidence Intervals
for Proportions: An Example
• If 22% of a random sample of 764 adult
Canadians smoke, what % of all adult
Canadians smoke?
– c.i. = .22 ±1.96 (√.25/764)
– c.i. = .22 ±1.96 (√.00033)
– c.i. = .22 ±1.96 (.018)
– c.i. = .22 ±.04
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6-20
Constructing Confidence Intervals
for Proportions: An Example (continued)
• Changing back to %’s, we can estimate
that 22% ± 4% of Canadian adults
smoke.
• Another way to state the interval:
– 18%≤Pu≤ 26%
– We estimate the population value is
greater than or equal to 18% and less than
or equal to 26%.
• This interval has a .05 chance of being
wrong.
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QUIZ
• In the smoking example above, can
you identify the following?
– Population
– Sample
– Statistic
– Parameter
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6-22
QUIZ
(continued)
• Population = All Canadian adults.
• Sample = the 764 people selected
for the sample and interviewed.
• Statistic = Ps = .22 (or 22%)
• Parameter = unknown. The % of all
adult Canadians who smoke.
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Controlling the Width of Confidence Intervals
Confidence interval widens as confidence level increases:
Confidence interval narrows as sample size increases:
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