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Confidence Interval for a Mean
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Confidence Interval for a Mean
Given
A random sample of size n from
a Normal population
or
a non Normal population where n is
sufficiently large.
A population at least 20 times the sample size n.
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Confidence Interval for a Mean
Result
A confidence interval is given by
X  E where
S
E t*
n
where t* is the appropriate critical value for the
T distribution with (n – 1) DF.
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Example
Rolls (single rolls) of paper leave a factory with
weights that are Normal with unknown mean.
n = 8 rolls are randomly selected
1494.5
1483.8
1512.3
1507.0
1503.6
1504.5
1495.4
1521.5
X  1502.83
S  11.62
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Example
Rolls (single rolls) of paper leave a factory with weights
that are Normal with unknown mean.
n = 8 rolls are randomly selected, their sample mean
weight is 1502.83 pounds; the sample standard
deviation is 11.62 pounds.
Determine a 95% confidence interval for the population
(or true) mean weight.
S
E t*
n
with (n  1)  (8  1)  7 DF
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Example
T7
t* = 2.365
0.025
0.95
0.025
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Example
Rolls (single rolls) of paper leave a factory with weights
that are Normal with unknown mean.
n = 8 rolls are randomly selected, their sample mean
weight is 1502.83 pounds; the sample standard
deviation is 11.62 pounds.
Determine a 95% confidence interval for the population
(or true) mean weight.
S
11.62
E t*
 2.365
 2.3654.108  9.72
n
8
X  E  1502.83  9.72
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Interpretation
1502.83  9.72
1502.83 – 9.72 = 1493.11
1502.83 + 9.72 = 1512.55
(1493.11, 1512.55)
95% of all samples* produce an interval that covers the
true mean . We have the interval from one sample,
chosen randomly.
We are 95% confident that the mean weight of all rolls
is between 1493.11 and 1512.55 pounds.
*Provided sampling is done from a Normal distribution. Because the sample is
small, if this requirement is not met, the confidence is not really 95%.
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TRUE / FALSE QUIZ
1. There’s a 95% confidence a roll is between
1493.11 and 1512.55 pounds.
1494.5
1483.8
1512.3
1507.0
1503.6
1504.5
1495.4
1521.5
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TRUE / FALSE QUIZ
1. There’s a 95% confidence a roll is between
1493.11 and 1512.55 pounds.
1494.5
1483.8
1512.3
1507.0
1503.6
1504.5
1495.4
1521.5
For this sample: 75%.
95% would be impossible (although closest
would be “all” – which is clearly not the case).
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TRUE / FALSE QUIZ
1. There’s a 95% confidence a roll is between
1493.11 and 1512.55 pounds.
FALSE: If you want to make this kind of statement,
construct a prediction interval. One way of forming
a 95% prediction interval is with the interval from
the 2.5th to the 97.5th percentiles. (For a data set with
8 observations, this is tough. There are other
ways…)
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Example
Rolls (single rolls) of paper leave a factory with
weights that are Normal with unknown mean.
n = 8 rolls are randomly selected
1494.5
1483.8
1512.3
1507.0
1503.6
1504.5
1495.4
1521.5
X  1502.83
S  11.62
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TRUE / FALSE QUIZ
2. There’s a 95% confidence the sample mean is
between 1493.11 and 1512.55 pounds.
X  1502.83
FALSE. The sample mean definitely is between
them; we can be 100% confident in that, because the
sample mean centers the interval.
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TRUE / FALSE QUIZ
3. There’s a 95% confidence another random
sample of rolls will have mean between 1493.11
and 1512.55 pounds.
FALSE: The confidence interval estimates the
population mean weight.
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TRUE / FALSE QUIZ
4. There’s a 95% probability that the mean
weight of all rolls is between 1493.11 and
1512.55 pounds.
FALSE (but in a way “closest” to true). The
probability is either 0 or 1 – depending on what the
population mean is.
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Interpretation
What “95% confidence means”
95% of all samples produce an interval that covers
the true mean .
We have an interval from one sample, chosen
randomly.
Our interval either does or does not cover : in
practice we just don’t know. We do know that the
procedure works 95% of the time.
Interpreting a 95% confidence interval
We are 95% confident that the mean weight of all
rolls is between 1493.11 and 1512.55 pounds.
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