  |
Average Movie Length (in minutes)
It's very difficult to get all movies in a data-base to
find the true mean movie length.
However, we can take a very reasonable sample of 30 movies and find the sample
mean is 108.6 minutes with a standard deviation of 12.4 minutes.
Find
the 99% confidence interval for the true mean movie length.
Solution:
E = (critical z-value)*(
)
= (
)*
()
E = ( 2.575 ) * (
)
= 5.8 minutes
99% confidence interval for true movie length is (108.6 - 5.8) < µ < (108.6 + 5.8)
or 102.8 minutes < µ < 114.4 minutes
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A random sample of 50 new
automobiles were chosen to
be documented for their total
vehicular weight. The mean of
this sample was 3358 pounds
with a standard deviation of
478 pounds.
Solution: 3225.5 pounds < µ < 3490.5 pounds
Find
the 95% confidence interval
for the true mean vehicular weight
The Central Limit Theorem (CLT) provides the basis for the calculations of
E.
We want (
- E) and then ( +
E) to get the likely range of values of the true population mean µ.
E = (critical z-value)*()
= ()*
()
E = ( 1.96 ) * (
)
= 132.5 pounds
95% confidence interval for true population mean
µ is:
(3358 - 132.5) pounds < µ < (3358 +132.5) pounds
We are 95% confident that the true mean is between those two end-points.
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A confidence interval is a range of values that is likely to contain the true value of the population parameter.
The degree of confidence is the probability that the confidence interval contains the true value of the population parameter.
A critical z-value is a boundary
value obtained from the standard normal distribution.
(Books on statistics will symbolize this with
where the subscript 
will
be explained below.)
In practice, statisticians decided that 99% , 98% , 95% and 90%
would be levels of confidence. These are the regions where the parameter is
likely to exist.
Math types use the percentage remaining: 1% , 2% , 5% and 10% as being equal
to 
(alpha).
That means that
percentage remains outside the interval. (on
each side.)
Critical z-values are found in the Standard
Normal Tables and can
be summarized in a
Mini-Table:
-Level
for each Tail
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|
In a previous chapter we mentioned that however useful the mean of a sample is to the analysis it is still better to present more details about this estimate.
The mean is one of the key parameters we study
in the introduction to statistics.
The mean of the sample ()
is one of our favourite estimators.
Now we present the calculations for how good an estimate the really
is.
We calculate a quantity called E which is the maximum error of the estimate.
This gives us (-
E ) and (
+ E) as boundaries for what we realize is the likely range of values of the
true mean of the population.
In other words the mean (µ ) of the population may never be known.
But our estimate ()
gives us an interval of values for µ.
Interval is (
+ E) < µ < (
- E) .