Transcript PARAMETRIC STATISTICAL INFERENCE

PARAMETRIC STATISTICAL INFERENCE
PARAMETER ESTIMATION
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Confidence intervals
Estimate a population parameter
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Mean
HYPOTHESIS TESTING
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To assess evidence provided by data about
some claim concerning a population
SIGNIFICANCE TESTING
2 ways of hypothesis testing – testing for
statistical significance:
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Classical hypothesis tests
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P-value method
REASONING OF TESTS OF SIGNIFICANCE
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An outcome that would rarely happen if a
claim were true is good evidence that the
claim is not true
PARAMETRIC STATISTICAL INFERENCE:
HYPOTHESIS TESTING
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Example 1: Suppose we wish to know
whether the mean number of weekly shopping
trips made by households in a particular
neighborhood of an urban area differs from the
mean value for the urban area as a whole
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Example 2: We measure carbon monoxide
concentration from different residential
neighborhoods in some city and we wish to
compare the levels (are levels in one area
significantly larger or smaller than levels
observed in the other areas)
Logic behind hypothesis testing:
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Examine the likelihood (probability) that a
alternate claim (or outcome) would occur if an
existing claim were true
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If the probability is high, then we say that there is
not enough evidence to accept the alternate claim
and we retain the original claim
If the probability is low, then we say that there is
enough evidence against the original claim in
support of the alternate claim
An outcome that would rarely happen if a claim
were true is good evidence that the claim is not
true
PARAMETRIC STATISTICAL INFERENCE:
HYPOTHESIS TESTING
Example: The mean household size in a certain city
is 3.2 persons with a standard deviation of =1.6. A
firm interested in estimating weekly household
expenditures on food takes a random sample of
n=100 households. To check whether the sample is
truly representative, the firm calculates the mean
household size of the sample to be 3.6 persons. What is
the probability the firm’s sample is not representative
of the city with respect to household size, such that
the sample mean is actually greater than the mean for the
entire city.
VOCABULARY OF HYPOTHESIS TESTS
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Statement of hypotheses
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Claims concern a population, so we express them
in terms of population parameters
What is the population parameter being tested?
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NULL HYPOTHESIS (Ho) – original claim - the
statement being tested in a statistical test
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Test is designed to assess the strength of evidence
against the null hypothesis
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Statement of “no effect” or “no difference”
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ALTERNATIVE HYPOTHESIS (Ha) – alternate claim
– the statement to the contrary of Ho
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3 possible combinations of Ho and Ha
(a) Ho: =o; Ha: o
(b) Ho: =o; Ha: >o
(c) Ho: =o; Ha: <o
PARAMETRIC STATISTICAL INFERENCE
More detail on hypothesis statements:
One-sided versus two-sided alternative
· One-sided: we are interested only in deviations
from the null hypothesis in one direction
· Two-sided: no direction is specified
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Example: State the null and alternate hypotheses for
each of the following problems:
1. The diameter of a spindle in a small motor is supposed
to be 5 mm. If the spindle is either too small or too
large, the motor will not work properly. The
manufacturer measures the diameter in a sample of
motors to determine whether the mean diameter has
moved away from the target.
2.
Census Bureau data show that the mean household
income in the area served by a shopping mall is
$52,000 per year. A market research firm questions
shoppers at the mall. The researchers suspect the
mean household income of mall shoppers to be higher
than that of the general population.