Download the mean of the sample means the standard deviation of sample mean

Section 6.5
The Central Limit
Theorem
Distribution of Sample Means
Population (with mean
µ and standard
deviation σ)
Distribution of
Sample Means
Mean of the Sample
Means
Normal
Normal (for any
sample size n)
x  
Normal
(approximately)
x  
Not normal
x  
Not normal with n > 30
Not normal with n ≤ 30
Standard Deviation of
the Sample Means
x 
x 
x 

n

n

n
Central Limit Theorem
Given:
1. The random variable x has a distribution (which may or may not
be normal) with mean µ and standard deviation σ.
2. Simple random samples all of size n are selected from the
population. (The samples are selected so that all possible samples
of the same size n have the same chance of being selected.)
Central Limit Theorem – cont.
Conclusions:
1. The distribution of sample x will, as the sample size
increases, approach a normal distribution.
2. The mean of the sample means is the population mean
µ.
3. The standard deviation of all sample means is

n
Practical Rules Commonly Used
1. For samples of size n larger than 30, the distribution of the
sample means can be approximated reasonably well by a
normal distribution. The approximation gets closer to a
normal distribution as the sample size n becomes larger.
2. If the original population is normally distributed, then for
any sample size n, the sample means will be normally
distributed (not just the values of n larger than 30).
Notation
the mean of the sample means
µx = µ
the standard deviation of sample mean

x = n
(often called the standard error of the mean)
Normal Distribution Example
As we proceed
from n = 1 to
n = 50, we see
that the
distribution of
sample means
is approaching
the shape of a
normal
distribution.
Uniform Distribution Example
As we proceed
from n = 1 to
n = 50, we see
that the
distribution of
sample means
is approaching
the shape of a
normal
distribution.
U-Shaped Distribution Example
As we proceed
from n = 1 to
n = 50, we see
that the
distribution of
sample means
is approaching
the shape of a
normal
distribution.
Important Point
As the sample size increases, the
sampling distribution of sample means
approaches a normal distribution.
Method for Finding Nonstandard Normal
Distribution Areas or Probabilities
(When being asked to find an area or probability you will use the same method)
In the following notations, represents a non-standardized sample of values.
P(a < x< b) : denotes the probability that the z score is between a and b.

To find this probability in your calculator, type: normalcdf(a, b, µ, )
n
a
b
P( x> a) : denotes the probability that the z score is greater than a.

To find this probability in your calculator, type: normalcdf(a, 99999, µ, )
n
a
P( x< a) : denotes the probability that the z score is less than a.

To find this probability in your calculator, type: normalcdf(–99999, a, µ, )
n
a
Example 1: Some passengers died when a water taxi sank in
Baltimore’s Inner Harbor. Men are typically heavier than
women and children, so when loading a water taxi, let’s
assume a worst-case scenario in which all passengers are
men. Based on data from the National Health and Nutrition
Examination Survey, assume that the population of weights
of men is normally distributed with µ = 172 lb and σ = 29 lb.
a) Find the probability that if an individual man is randomly
selected, his weight is greater than 175 lb.
Example 1: Use the Chapter Problem from page 249 of your
textbook. It noted that some passengers died when a water
taxi sank in Baltimore’s Inner Harbor. Men are typically
heavier than women and children, so when loading a water
taxi, let’s assume a worst-case scenario in which all
passengers are men. Based on data from the National Health
and Nutrition Examination Survey, assume that the
population of weights of men is normally distributed with µ =
172 lb and σ = 29 lb.
b) Find the probability that 20 randomly selected men will
have a mean weight that is greater than 175 lb (so that their
total weight exceeds the safe capacity of 3500 pounds).
Example 2: Cans of regular Coke are labeled to
indicate that they contain 12 oz. Data Set 17 in
Appendix B lists measured amounts for a sample of
Coke cans. The corresponding sample statistics are n =
36 and x = 12.19 oz. If the Coke cans are filled so that µ
= 12.00 oz (as labeled) and the population standard
deviation is σ = 0.11 oz (based on the sample results),
find the probability that a sample of 36 cans will have
a mean of 12.19 oz or greater.
Example 3: When women were allowed to become
pilots of fighter jets, engineers needed to redesign the
ejection seats because they had originally been
designed for men only. The ACES-II ejection seats
were designed for men weighing between 140 lb and
and 211 lb. The weights of women have a mean of
143 lb and a standard deviation of 29. (based on data
from the National Health Survey.) If 36 different
women are randomly selected, find the probability
that their mean weight is between 140 lb and 211 lb.