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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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