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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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