Transcript What if the Original Distribution Is Not Normal? - Milan C

What if the Original
Distribution Is Not Normal?
Use the Central Limit Theorem.
Central Limit Theorem
If x has any distribution with mean  and
standard deviation , then the sample
mean based on a random sample of size n
will have a distribution that approaches the
normal distribution (with mean  and
standard deviation  divided by the square
root of n) as n increases without bound.
How large should the sample
size be to permit the
application of the Central Limit
Theorem?
In most cases a sample size of
n = 30 or more assures that the
distribution will be approximately
normal and the theorem will apply.
Central Limit Theorem
Central Limit Theorem
• For most x distributions, if we use a
sample size of 30 or larger, the
distribution will be approximatelyx normal.
Central Limit Theorem
• The mean of the sampling distribution is
the same as the mean of the original
distribution.
• The standard deviation of the sampling
distribution is equal to the standard
deviation of the original distribution divided
by the square root of the sample size.
Central Limit Theorem Formula
x  
Central Limit Theorem
Formula

x 
n
Central Limit Theorem
Formula
z 
x  

x
x  

 / n
x
Application of the Central Limit
Theorem
Records indicate that the packages shipped by a
certain trucking company have a mean weight of
510 pounds and a standard deviation of 90
pounds. One hundred packages are being shipped
today. What is the probability that their mean
weight will be:
a.
b.
c.
more than 530 pounds?
less than 500 pounds?
between 495 and 515 pounds?
Are we authorized to use the
Normal Distribution?
Yes, we are attempting to
draw conclusions about
means of large samples.
Applying the Central Limit
Theorem
What is the probability that their mean weight will
be more than 530 pounds?
Consider the distribution of sample means:
 x  510,  x  90 / 100  9
P( x > 530): z = 530 – 510 = 20 = 2.22
9
9
.0132
P(z > 2.22) = _______
Applying the Central Limit
Theorem
What is the probability that their mean weight
will be less than 500 pounds?
P( x < 500): z = 500 – 510 = –10 = – 1.11
9
9
.1335
P(z < – 1.11) = _______
Applying the Central Limit
Theorem
What is the probability that their mean weight
will be between 495 and 515 pounds?
P(495 < x < 515) :
for 495: z = 495 – 510 =  15 =  1.67
9
9
for 515: z = 515 – 510 = 5 = 0.56
9
9
.6648
P(  1.67 < z < 0.56) = _______