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to estimate the population mean, this is called ‘‘inferential statistics’’ because we
are inferring the population mean from the sample mean.
When we study a sample of people in science research, we know the size of our
sample (n), the mean of our sample (X), and the standard deviation of our sample
(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 learn more about this test from studying any good statistics textbook (e.g.
Levine ( 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 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 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
book, 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
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