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We are 95 % conﬁdent that the population mean for the Chevy Impala is somewhere
between 26.99 and 28.67 mpg.
Based upon the 28 mpg of the Chevy Impala we could create a billboard
emphasizing the higher miles per gallon and highlight a perceived lower
environmental impact. Our data supports this claim because the 28 mpg is inside of
this 95 % conﬁdence interval for the population mean.
You are probably asking yourself: ‘‘Where did that 1.96 in the formula come
from?’’
3.1.4 Where Did the Number ‘‘1.96’’ Come From?
A detailed mathematical answer to that question is beyond the scope of this topic,
but here is the basic idea.
We make an assumption that the data in the population are ‘‘normally
distributed’’ in the sense that the population data would take the shape of a ‘‘normal
curve’’ if we could test all of the people in the population. The normal curve looks
like the outline of the Liberty Bell that sits in front of Independence Hall in
Philadelphia, Pennsylvania. The normal curve is ‘‘symmetric’’ in the sense that if
we cut it down the middle, and folded it over to one side, the half that we folded
over would ﬁt perfectly onto the half on the other side.
A discussion of integral calculus is beyond the scope of this topic, but
essentially we want to ﬁnd the lower limit and the upper limit of the population data in
the normal curve so that 95 % of the area under this curve is between these two
limits. If we have more than 40 people in our research study, the value of these
limits is plus or minus 1.96 times the standard error of the mean (s.e.) of our
sample. The number 1.96 times the s.e. of our sample gives us the upper limit and
idea, you can consult a good statistics book (e.g. Salkind 2010 or van Emden
2008 ).
The number 1.96 would change if we wanted to be conﬁdent of our results at a
different level from 95 % as long as we have more than 40 people in our research
study.
For example:
1. If we wanted to be 80 % conﬁdent of our results, this number would be 1.282.
2. If we wanted to be 90 % conﬁdent of our results, this number would be 1.645.
3. If we wanted to be 99 % conﬁdent of our results, this number would be 2.576.
But since we always want to be 95 % conﬁdent of our results in this topic, we
will always use 1.96 in this topic whenever we have more than 40 people in our
research study.
By now, you are probably asking yourself: ‘‘Is this number in the conﬁdence
interval about the mean always 1.96?’’ The answer is: ‘‘No!’’, and we will explain
why this is true now.
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