Transcript 6.3 One and Two-Sample Inference for Means

6.3 One and Two-Sample
Inference for Means
Example
JL Kim did some crude tensile strength testing on pieces of some
nominally .012 in diameter wire of various lengths. The lengths are
given in the data file WIRE in the class directory.
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If one is to make a confidence interval for the mean measured
strength of 25 pieces of this wire, what model assumption must be
employed? Make a plot useful in assessing the reasonableness of
this assumption.
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Make a 95% two-sided confidence interval for the mean measured
strength of the 25cm pieces of this wire.
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Is there sufficient evidence to suggest that the diameter differs from
nominal?
Overview
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Previously we have discussed ways of making
inferential statements concerning a population mean
when the population standard deviation is known
and the sample size is large
Actually, one will rarely know what the true
population standard deviation is and will have to
approximate the population standard deviation using
the collected sample
Furthermore, one will not always be able to collect a
large sample due to various physical constraints
If n is large, then use the previous methods
Small Sample Inference for µ
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(Student) t distribution should be used
Table B.4 on page 790 gives Quantiles
Degrees of freedom = ν = “nu”= n-1
Probability densities of t are bell-shaped and
symmetric about zero
Flatter than standard normal density but are
increasingly like it as ν gets larger
For ν >30, t and z distributions are almost
indistinguishable
Test Statistic
T
x
s
n
Example
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Part of a data set of W. Armstrong gives
numbers of cycles to failure of 10 springs of a
particular type under a stress of 950 N/mm2.
These spring-life observations are given
below in units of 1,000 cycles.
225,171,198,189,189,135,162,135,117,162
What is the average spring lifetime under
these conditions?
Confidence Interval
x t*
s
n
Minitab Output
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Descriptive Statistics: lieftime
Variable
N
Mean
StDev
lieftime
10
168.3
33.1
SE Mean
10.5
Example – part B
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An investigator claims that the lifetime is less
than 190,000 cycles. Test his claim at α = .05.
Example 2
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A cereal company claims that the mean
weight of cereal in a “16-ounce box” is at
least 16.15 ounces. As a means of testing the
company’s claim, a consumer advocacy group
randomly samples 10 boxes and obtains an
average weight of 16.06 ounces and a
standard deviation of .10 oz. Conduct the
appropriate hypothesis test using a
significance level of 0.05.