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
A1.1 Sampling Distribution of the Sample Mean: Normal Population
Example 1: IQ is a measurement of intelligence derived from the Stanford Binet IQ test. Scores on this test
are normally distributed with a mean score of 100 and a standard deviation of 15.
(a) (i)
Use StatCrunch to simulate 1000 random samples of size n  10 from this population.
StatCrunch -->StatCrunch Website -->Open StatCrunch --> Data --> Simulate --> Normal
Click Options and Save.
Explain the representation of entries in row 1 of the StatCrunch spreadsheet.
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(ii) Use StatCrunch to calculate the sample mean for each random sample of size n  10 .
Stat --> Summary Stats --> Rows ---> Select all Normal1 to Normal10 for Select column(s)
--> Select Mean for Statistics --> Check Store in data table --> Compute --> Options ---> Save.
Explain the representation of entries in column 11 of the StatCrunch spreadsheet.
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(iii) Use StatCrunch to draw a histogram for the 1000 sample means.
What is the sampling distribution of the sample means of sample size n = 10?
Let's use a lower class limit of the first class of 65 and a class width of 5.
Graph --> Histogram --> Select Row Mean for Select column(s) --> Input Start at: 65
and Width: 5 for Bins ---> Compute ---> Options ---> Save.
Sketch the histogram and comment on the shape of the distribution.
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(b) Use StatCrunch to find the mean and standard deviation of the sampling distribution of
the 1000 sample means?
Stat ---> Summary Stats ----> Columns ---> Select Row Mean ---> Select Mean and Std. dev.
for Statistics ---> Compute ---> Options ---> Save.
Write down the results.
 x = ___________
 x = ___________
(c) Repeat part (a) and (b) with size n  40 .
Sketch the histogram and comment on the shape of the distribution.
Write down the results for  x = ___________ and  x = ___________
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A1. Sampling Distributions of Mean
A1.2 Sampling Distribution of the Sample Mean: Nonnormal Population
Example 1: The waiting time in line can be modeled by an exponential distribution which is similar to skewed to
the right with a mean of 5 minutes and a standard deviation of 5 minutes.
(a) Repeat Example 1 of A1.1 or part (a) but using exponential distribution instead of normal distribution.
StatCrunch -->StatCrunch Website -->Open StatCrunch --> Data --> Simulate --> Exponential,
then follow the steps given in (i) to (iii) to construct a histogram. If you are not sure about the start
at value and the class width, just leave them blank and StatCrunch will figure it out based on
the 1000 sample means.
Sketch the histogram and comment on the shape of the distribution.
(b) Repeat Example 1 of A1.1 of part (b) using the sample means obtained from A1.2 of part (a).
Write down the results.
 x = ___________
 x = ___________
(c) Repeat part (a) and (b) with size n  40 .
Sketch the histogram and comment on the shape of the distribution.
Write down the results for  x = ___________ and  x = ___________
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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 .
Use the distribution shapes obtained from the two simulation Example 1 results of objective A1
to verify the statement A of the Central Limit Theorem.
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 where  x   .
The standard deviation ofthe sampling distribution of x is  x where  x 

n.
Use the  x and  x obtained from the two simulation Example 1 of objective A1 to verify
the statement B of the Central Limit Theorem.
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?
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(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 : Standardize x to Z :
Example 1 :
Z
x

x  x
x
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) Find  x and  x .
(c) Use StatCrunch to find P( x  62.6) ?
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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) Use StatCrunch to 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?
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Ch 8.2 Distribution of the Sample Proportion
Objective A : Shape, Center and Spread of the Distribution of p̂ .
Distribution of the Sample Proportions - Explain what is a p̂ distribution.
x
.
n
The shape of the sampling distribution of p̂ is approximately normally provided by,
A. Sampling distribution of sample proportion p̂ , where pˆ 
np(1  p)  10 or npq  10 where q  1  p .
B. Finding the mean and standard deviation of p̂
 pˆ  p
,
 pˆ 
p(1  p)
n
Objective B : Finding Probability of p̂ that is Normally Distributed
Standardize p̂ to Z
Z
pˆ   pˆ
 pˆ
where  pˆ  p and  pˆ 
p(1  p)
n
provided p̂ 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 p̂ .
(b) If 154 college students in the sample send and receive text messages on the cell phone,
find p̂ ,  p̂ , and  p̂
(c) Use StatCrunch to find 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?
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Example 2: According to creditcard.com, 29% of adults do not own a credit card.
(a) Suppose a random sample of 500 adults is asked, "Do you own a credit card?"
Describe the sampling distribution of p̂ , the proportion of adults who own a credit card.
(b) Use StatCrunch to find the probability that in a random sample of 500 adults between
25% and 30% do not own a credit card?
(c) Would it be unusual for a random sample of 500 adults to result in 125 or fewer who
do not own a credit card? Why?
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