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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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