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from it 1.96 times the standard error of the mean to get the lower limit of the conﬁdence
interval.
You will remember (See Sect. 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.
In mathematical terms, the formula for the 95 % conﬁdence interval about the
mean is:
X 1 : 96 s : e :
ð 3 : 1 Þ
Note that the ‘‘± sign’’ stands for ‘‘plus or minus,’’ and this means that you ﬁrst
add 1.96 times the s.e. to the mean to get the upper limit of the conﬁdence interval,
and then subtract 1.96 times the s.e. from the mean to get the lower limit of the
conﬁdence interval. Also, the symbol 1.96 s.e. means that you multiply 1.96 times
the standard error of the mean to get this part of the formula for the conﬁdence
interval.
Note: We will explain shortly where the number 1.96 came from.
Let’s try a simple example to illustrate this formula.
3.1.3 Estimating the Conﬁdence Interval for the Chevy Impala
in Miles Per Gallon
Let’s suppose that you have been asked to be a part of a larger study looking at the
carbon footprint of Chevy Impala drivers. You are interested in the average miles
per gallon (mpg) of a Chevy Impala. You asked owners of the Chevy Impala to
keep track of their mileage and the number of gallons used for two tanks of gas.
Let’s suppose that 49 owners did this, and that they average 27.83 miles per gallon
(mpg) with a standard deviation of 3.01 mpg. The standard error (s.e.) would be
3.01 divided by the square root of 49 (i.e., 7) which gives a s.e. equal to 0.43.
The 95 % conﬁdence interval for these data would be:
27 : 83 1 : 96 ð 0 : 43 Þ
The upper limit of this conﬁdence interval uses the plus sign of the ± sign in
the formula. Therefore, the upper limit would be:
27 : 83 þ 1 : 96 0 : 43
ð
Þ ¼ 27 : 83 þ 0 : 84 ¼ 28 : 67 mpg
Similarly, the lower limit of this conﬁdence interval uses the minus sign of
the ± sign in the formula. Therefore, the lower limit would be:
27 : 83 1 : 96 ð 0 : 43 Þ¼ 27 : 83 0 : 84 ¼ 26 : 99 mpg
The result of our part of the ongoing research study would, therefore, be the
following:
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