Transcript Chapter 6: Introduction to Formal Statistical

Chapter 6: Introduction to Formal
Statistical Inference
November 19, 2008
6.2 Large Sample Tests for a Mean
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While a confidence interval is an inferential
technique that attempts to estimate a
population parameter, a test of significance
is an inferential technique used to determine
the validity of some claim concerning a
population based on sample data
The reasoning behind tests of significance is
based on what would happen over repeated
sampling
The Basics
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Some claim is made concerning the population of
interest.
As a means of determining the validity of the claim, a
sample is collected and the appropriate statistics and
sampling distributions are obtained.
Assuming that the claim is true, one will determine
how likely the obtained sample results are.
If one concludes that the obtained sample data are
not likely to be obtained given the claim is true, then
one will conclude the claim is false.
The Hypotheses
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Null Hypothesis: statement of the form
Parameter = #
Forms the basis of investigation in a
significance test
Usually formed to embody a status quo/ “predata” view of the parameter
Denoted as Ho (H naught)
The Hypotheses
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Alternative Hypothesis is a statement that
stands in opposition to the null hypothesis.
Specifies what forms of departure from the
null hypothesis are of concern (>, < or ≠)
Denoted as Ha
The Hypotheses for a Test of Mean
Three possible pairs of hypotheses:
Ho: µ = #
Ho: µ = #
Ho: µ = #
Ha: µ > #
Ha: µ < #
Ha: µ ≠ #
Test Statistic
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The particular form of numerical data
summarization used in a significance test.
The formula typically involves the number
appearing in the null hypothesis.
The reference distribution for the test statistic
is the probability distribution describing the
test statistic, provided the null hypothesis is
in fact true.
Test Statistic for Large Sample Test
where σ is Known
z
x #
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n
Step 3: p value
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The observed level of significance (p-value)
is the probability that the reference
distribution assigns to the set of possible
values of the test statistic that are at least as
extreme as the one actually observed
In terms of casting doubt on the null
hypothesis, small p-values are evidence
against Ho
5 Step Format for Hypothesis Tests
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State the null hypothesis
State the alternative hypothesis
State the test criteria (test statistic and
reference distribution)
Show the sample-based calculations.
Report a p-value and state its implications
in the context of the problem
Example: Baby Food
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Suppose the declared label weight of the baby food
is 135 grams and process engineers have set a
target mean net fill weight of 139.8 grams. (Given
that σ = 1.6 grams) Suppose that in a routine check
of filling-process performance, intended to detect
any change of the process mean from its target
value, a sample of 25 jars produced an average of
139.0 g. What does this value have to say about the
plausibility of the current process mean actually
being at the target of 139.8 grams?
Generally Applicable Large-n
Significance Tests for µ
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For observations that
are describable as
essentially equivalent to
random selections with
replacement from a
single population with
mean µ and variance
σ2, if n is large
z
x #
s
n
Example
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A vendor claims that bottles produced by his
manufacturing process have a mean internal
strength of 150 psi. A potential customer thinks that
the vendor is overstating the strength and selects a
random sample of 36 bottles from the line. These
bottles have a mean internal strength of 148 psi
with a standard deviation of 5.5 psi. Is there
enough evidence to refute the vendor’s claim?