Download Chapter 9: Inference of One Population Mean

Chapter 9: Inference of One
Population Mean
9.2 The t-distribution
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The t-distribution is bell shaped and
symmetric about 0
The probabilities depend on the
degrees of freedom, df
The t-distribution has thicker tails and
is more spread out than the standard
normal distribution
t-Distribution (A.2)
t  critical value, t - score for two - tail : t( ,n 1)
t(.05, 4 )  2.776, t(.10, 4 )  2.132
9.3 Confidence Interval for a Population Mean
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Point estimate ± margin of error
Parameter: population mean 
Point estimate: the sample mean x
The exact standard error: σ/ n
In practice, we estimate σ by the sample
standard deviation, s
How to Construct a Confidence
Interval for a Population Mean
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For large n, use CLT
For small n from an underlying population
that is normal…
The confidence interval for the population
mean is:
x  z*

n
Confidence Interval for , known 
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95% confidence interval
( LL0.95  x  1.96 * x ,UL0.95  x  1.96 * x )
• Bluegill sunfish example:
• LL0.95=159.40-1.96*6.21=147.23
• UL0.95=159.40+1.96*6.21=171.57
• Thus we are 95% confident that (147,23, 171,57)
contains the population mean.
Common used confidence levels
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Remark: For different confidence level,
use the corresponding z-value.
95%: z=1.96
99%: z=2.576
Confidence Interval for µ,
unknown 
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In practice, we don’t know the
population standard deviation
Substituting the sample standard
deviation s for σ to get se = s/ n
introduces extra error
To account for this increased error,
we replace the z-score by a slightly
larger score, the t-score
9.3 Confidence Interval for ,
unknown 
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In practice, we estimate the standard
error of the sample mean by se = s/ n
Then, we multiply se by a t-score from
the t-distribution to get the margin of
error
95% Confidence Interval for 
unknown 
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A 95% confidence interval for µ is:
s
x  t (.05, n -1) ( )
n
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To use this method, you need:
•
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Data obtained by randomization
An approximately normal population distribution
Example: Mosquito fish length
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A random sample of size 20 male
mosquito fish gives the sample mean
21.0mm with the sample standard
deviation 1.76mm
Construct 95% CI for the mean length
Assume the length is approximately
normal distributed.
Solution
s
1.76

 0.394
n
20
 2.093
n  20, s x 
t( 0.05,19 )
me  2.093 * 0.394  0.825
LL.95  x  me  21.0  0.825  20.175
UL.95  x  me  21.0  0.825  21.825
95% CI for  is
21.0  0.825  (20.175, 21.825)
We conclude with 95% confident that
the range of 20.175mm to 21.825mm includes
the true mean lengh of mosquito fish.
How Do We Find a t- Confidence
Interval for Other Confidence
Levels?
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The 95% confidence interval uses
t(.05,n-1) since 95% of the probability
falls between - t(.05, n-1) and t(.05,n-1)
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For 99% confidence, the error
probability is 0.01 and the appropriate
t-score is t(.01,n-1).
If the Population is Not Normal,
is the Method “Robust”?
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A basic assumption of the confidence
interval using the t-distribution is that
the population distribution is normal
Many variables have distributions that
are far from normal
If the Population is Not Normal,
is the Method “Robust”?
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Question: How problematic is it if we
use the t- confidence interval even if the
population distribution is not normal?
Answer:
• For large random samples, it’s not
•
problematic
The Central Limit Theorem applies: for large
n, the sampling distribution is bell-shaped
even when the population is not
If the Population is Not Normal,
is the Method “Robust”?
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What about a confidence interval using the
t-distribution when n is small?
Even if the population distribution is not
normal, confidence intervals using t-scores
usually work quite well
We say the t-distribution is a robust method
in terms of the normality assumption
Cases Where the t- Confidence
Interval Does Not Work
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With binary data
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With data that contain extreme outliers
The Standard Normal Distribution is
the t-Distribution with df = ∞
9.4 Reporting a sample mean
Three ways of reporting：
(i ) x  s
（ii ) x  s x
(iii ) x  me with confidence level
Graphic representation