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In mathematical terms, the formula for the 95% conﬁdence interval about the

mean is:

X

1

:

96 s

:

e

:

(3.1)

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.

Note that the
“

3.1.3 Estimating the Conﬁdence Interval 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 (0.43)

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)

═

27.83

þ

0.84

═

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 (0.43)

═

27.83

─

0.84

═

26.99 mpg

The result of our part of the ongoing research study would, therefore, be the

following:

“We are 95% conﬁdent that the population mean for the Chevy Impala is somewhere

between 26.99 mpg 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

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