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3.25 ECO 3411 31 Central Limit Theorem
• In selecting a random sample of size n from a population,
−
the sampling distribution of the sample mean x can be
approximated by a normal probability distribution as the
sample size becomes large
−
• The sampling distribution of x can be approximated by
a normal probability distribution whenever the sample size
is large. The largesample condition can be assumed for
simple random samples of size 30 or more
• Whenever the population has a normal probability
−
distribution, the sampling distribution of x has a normal
probability distribution for any sample size
• Please go to: www.ruf.rice.edu./~lane/rvls.html
ECO 3411 32 Sampling Distribution of x
• If we use a large (n > 30) simple random sample, the
central limit theorem enables us to conclude that the
sampling distribution of x can be approximated by a
normal probability distribution.
• When the simple random sample is small (n < 30), the
sampling distribution of x can be considered normal only
if we assume the population has a normal probability
distribution. ECO 3411 33 Example: UCF
• Sampling Distribution of x for the SAT Scores σ
80
σx =
=
= 14.6
n
30 E ( x ) = µ = 990 ECO 3411 x 34 Example: UCF BUSINESS STUDENTS
• Sampling Distribution of x for the SAT Scores
What is the probability that a simple random sample
of 30 applicants will provide an estimate of the population
mean SAT score that is within plus or minus 10 of the
actual population mean µ ?
In other words, what is the probability that x will be
between 980 and 1000? ECO 3411 35 Example: UCF
• Sampling Distribution of x for the SAT Scores
Sampling distribution
of x
Area = .2518 Area = .2518 x 980 990 1000
Using the standard normal probability table with
z = 10/14.6= .68, we have area = (.2518)(2) = .5036
ECO 3411 36 Relationship Between the Sample Size and the
x
Sampling Distribution of Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. E( ) = µ regardless of the...
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This note was uploaded on 01/17/2014 for the course ECO 3411 taught by Professor Staff during the Winter '08 term at University of Central Florida.
 Winter '08
 Staff

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