Module 16: One-sample t-tests and Confidence Intervals

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Module 16 One-sample t-tests and Confidence
Intervals
This module presents a useful statistical tool,
the one-sample t-test and the confidence interval
for the population mean.
Reviewed 05 May 05 / Module 16
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The t-test and t Distribution
What happens when we don't know the true value
for the population standard deviation ?? Suppose
we have only the information from a random
sample, n 5, from the population of body
weights, with 153.0 lbs, and s 12.9
lbs. When we had a value for the population
parameter ?, we used the following formula
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t-test and t distribution (contd.)
We do have an estimate of the population standard
deviation, ?, namely the sample standard
deviation s 12.9 lbs. Hence, it seems
reasonable to think that we should be able to use
this estimate in some way. It is also reasonable
to think that, if we substitute s for ?, we are
substituting a guess at the truth for the truth
itself and we will probably have to pay a price
for doing so. So, what is the price? The
essence of the situation is that, when we
substitute a guess for the truth, we add noise to
the system. The question then becomes one of
characterizing this noise and taking it into
account. Noise in this situation is equivalent
to variability, so we are adding variability to
the system. How much and exactly where?
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t-test and t distribution (contd.)
If we use this estimate, then we must make
appropriate adjustments to the formula to account
for the variability of this estimate.   To
properly account for this situation, we need to
use a distribution different from the normal
distribution. The appropriate distribution is
the t distribution, which is very similar to the
normal distribution for large sample sizes, but
differs importantly for smaller samples,
especially those with n lt 30.
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Confidence Interval for µ using s
The appropriate formula, when ? 0.05,
is   where t0.975(n-1) references the t
distribution with n-1 degrees of freedom (df),
specifically the point on that distribution below
which lies 0.975 of the total area. For this
situation, the correct number of degrees of
freedom is one less that the sample size, i.e. df
n -1.
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Tables for the t Distribution
To obtain the values for the t distribution, see
Table Module 2 The t distribution.
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Example
Given this confidence interval, would you believe
that the population mean for the population from
which this sample was selected had the value ?
170.0 lbs?
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Ten Samples from N(150,10), n 5
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95 Confidence Intervals for samples n 5
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Ten Samples from N(150,10), n 20
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95 Confidence Intervals for samples n 20
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Ten Samples from N(150,10), n 50
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95 Confidence Intervals for samples n 50
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