Chapter 8 Sampling Distribution
Ch 8.1 Distribution of Sample Mean
Objective A : Shape, Center, and Spread of the Distributions of x
A1. Sampling Distributions of Mean
Assume equal chances for each number to be selected.
S  {1, 2,3}
Sampling Probability distribution of mean x .
Let’s say we select two elements (n  2) from S  {1, 2,3} with replacement. (Independent case)
List out all possible combinations (sample space) and x for each combination.
Probability distribution of x is summarized in the table shown below.
Probability histogram for x .
Let’s compare the distribution shape of x and x .
P( x)
1
3
1
2
1
1.5 2
3
x
P( x)
39
29
19
2.5 3
x
Is this by chance that x is normally distributed?
A2. Central Limit Theorem
A. If the population distribution of x is normally distributed, the sampling distribution of x is normally
distributed regardless of the sample size n .
If the population distribution x is not normally distributed, the sampling distribution of x is
guaranteed to be normally distributed if n  30 .
B. Mean/standard deviation of a sampling distribution of x vs mean/standard deviation of a
population distribution of x .
The mean and standard deviation of population distribution are  and  respectively.
The mean of the sampling distribution of x is  x .
x  
The standard deviation of the sampling distribution of x is  x .
x 

n
Example 1 : Determine  x and  x from the given parameters of the population and the sample
size.
  27 ,   6 , n  15
Example 2 : A simple random sample is obtained from a population with   64 and   18 .
(a) If the population distribution is skewed to the right, what condition must be applied in
order to guarantee the sampling distribution of x is normally distributed?
(b) If the sample size is n  9 , what must be true regarding the distribution of the
population in order to guarantee the sampling distribution of x to be normally
distributed?
Objective B : Finding Probability of x that is Normally Distributed
Standardize x to Z
Recall : Standardize x to Z :
Z
Now :
Z
Standardize x to Z :
x

x  x
x
Example 1 : A simple random sample of size n  36 is obtained from a population mean
  64 and population standard deviation   18 .
(a) Describe the sampling distribution x .
(b) What is P( x  62.6) ?
Example 2 : The upper leg of 20 to 29 year old males is normally distributed with a mean
length of 43.7cm and a standard deviation of 4.2cm.
(a) What is the probability that a random sample of 12 males who are 20 to 29 years
old results in a mean upper leg length that is between 42cm and 48cm?
(b) A random sample of 15 males who are 20 to 29 years old results in a mean upper
leg length greater than 46 cm. Do you find the result unusual? Why?
Ch 8.2 Distribution of the Sample Proportion
Objective A : Shape, Center and Spread of the Distribution of p
Distribution of the Sample Proportions
A. Sampling distribution of sample proportion p , where. p 
x
n
The shape of the sampling distribution of p is approximately normally provided by,
np(1  p)  10
or npq  10 , where q  1  p
B. Finding the mean and standard deviation of p
p  p
p 
p(1  p)
n
Objective B : Finding Probability of p that is Normally Distributed
Standardize p to Z
Z
p  p
p
where  p  p and  p 
p(1  p)
n
provided p is approximately normally distributed.
Example 1: A nationwide study indicated that 80% of college students who use a cell phone,
send and receive text messages on their phone. A simple random sample of
n  200 college students using a cell phone is obtained.
(a) Describe sampling distribution of p .
(b) What is the probability that 154 or fewer college students in the sample send
and receive text messages on the cell phone? Is this unusual?