Download Section 6.1

Section 6.1
Confidence Intervals
for the Mean
(large samples)
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The Energy Policy Conservation Act (EPCA) was
enacted in 1975. This act established the
Corporate Average Fuel Economy (CAFÉ)
standards for passenger cars and light trucks.
These standards require that the fuel economy
rating for a manufacturer’s entire line of
passengers cars must average at least 27.5 miles
per gallon, and the fuel economy rating for a
manufacturer’s line of light trucks must average at
least 20.7 miles per gallon
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Automobile manufactures use the descriptive statistics to
analyze the data collected during vehicle tests conducted
in their laboratories.
TO meet the CAFÉ requirements automobile manufactures test
the mean fuel economy rating of preproduction prototype of
their new vehicles and submit their test results to the EPA
(Environmental Protection Agency). The EPA retests the
mean fuel economy rating of a sample of the new vehicles,
about 10% of the tested vehicles to confirm the
manufacturer’s results. In a recent year the EPA tested
sample of 73 cars from an automobile manufacturer’s line of
passenger cars. The mean fuel economy rating was 28.8
miles per gallon.
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−
x = 28.8mpg
The problem with the point estimate is that it is rarely equal
to the exact parameter of the population.
To make more meaningful estimate by specifying an
interval of values, together with a statement of how
confident we are that your interval estimate contains the
population parameter.
Lets suppose that the EPA wants to be 90% confident of its
estimate of the mean fuel economy rating for the
manufacturer’s entire line of passenger cars.
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Point Estimate
DEFINITION:
A point estimate is a single value
estimate for a population parameter.
The best point estimate of the
population mean
is the sample mean
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Interval Estimates
Point estimate
28.8•
An interval estimate is an interval or range of
values used to estimate a population parameter.
(
28.8•
27.7
)
29.9
The level of confidence, x, is the probability that
the interval estimate contains the population
parameter.
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Distribution of Sample Means
When the sample size is at least 30, the sampling
distribution for
is normal.
Sampling distribution of
For c = 0.90
0.05
0.95
0.05
-1.645 0 1.645
z
90% of all sample means will have standard
scores between z = -1.645 and z = 1.645
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Maximum Error of Estimate
OR Margin of Error
The maximum error of estimate E is the greatest possible distance
between the point estimate and the value of the parameter it is,
estimating for a given
level of confidence, c.
When n
used for
30, the sample standard deviation, s, can be
.
Find E , the maximum error of estimate or M the Margin of
error using s = 5.71 mpg
Using zc = 1.645, s = 5.71, and
M = zc
n = 73,
 5 . 71 

= 1 . 645 * 
n
 73 
s
You are 90% confident that 8the margin of error is 1.1mpg
Confidence Intervals for
Definition: A c-confidence interval for the population mean is
−
x −
M
<
µ
<
−
x +
M
Find the 90% confidence interval for the mean fuel economy
rating for the manufacturer’s entire line of passenger cars
You found
Left endpoint
(
27.7
= 28.8 and M = 1.1
Right endpoint
28.8•
)
29.9
With 90% confidence, you can say that the mean fuel
economy rating for the manufacturer’s
entire line of cars is
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between 27.7 and 29.9 mpg
Sample Size
Given a c-confidence level and an maximum error of estimate,
E, the minimum sample size n, needed to estimate , the
population mean is
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