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(STDEV). We use these ﬁgures to estimate the population mean with a test called
the “conﬁdence interval about the mean.”
3.1.2 Estimating the Lower Limit and the Upper Limit of the 95
Percent Conﬁdence Interval About the Mean
The theoretical background of this test is beyond the scope of this topic, and you can
(2011) or Bremer and Doerge (2010)), but the basic ideas are as follows.
We assume that the population mean is somewhere in an interval which has a
“lower limit” and an “upper limit” to it. We also assume in this topic that we want to
be “95% conﬁdent” that the population mean is inside this interval somewhere. So,
we intend to make the following type of statement:
“We are 95% conﬁdent that the population mean in miles per gallon (mpg) for the Chevy
Impala automobile is between 26.92 miles per gallon and 29.42 miles per gallon.”
If we want to create a billboard emphasing the perceived lower environmental
impact of the Chevy Impala by claiming that this car gets 28 miles per gallon (mpg),
we can do this because 28 is inside the 95% conﬁdence interval in our research
study in the above example. We do not know exactly what the population mean is,
only that it is somewhere between 26.92 mpg and 29.42 mpg, and 28 is inside this
interval.
But we are only 95% conﬁdent that the population mean is inside this interval,
and 5% of the time we will be wrong in assuming that the population mean is 28
mpg.
But, for our purposes in science research, we are happy to be 95% conﬁdent that
our assumption is accurate. We should also point out that 95% is an arbitrary level
of conﬁdence for our results. We could choose to be 80% conﬁdent, or 90%
conﬁdent, or even 99% conﬁdent in our results if we wanted to do that. But, in
this topic, we will always assume that we want to be 95% conﬁdent of our results.
That way, you will not have to guess on how conﬁdent you want to be in any of the
problems in this topic. We will always want to be 95% conﬁdent of our results in
this topic.
So how do we ﬁnd the 95% conﬁdence interval about the mean for our data?
In words, we will ﬁnd this interval this way:
“Take the sample mean ( X ), and add to it 1.96 times the standard error of the mean (s.e.) to
get the upper limit of the conﬁdence interval. Then, take the sample mean, and subtract
from it 1.96 times the standard error of the mean to get the lower limit of the conﬁdence
interval.”
You will remember (See Section 1.3 )that the standard error of the mean (s.e.) is
found by dividing the standard deviation of our sample (STDEV) by the square root
of our sample size, n.
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