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STA 2023
Chapter 6 – Sampling Distributions

What is a Sampling Distribution? (6.1) – SKIP

Properties of Sampling Distributions: Unbiasedness and Minimum Variance (6.2) –
SKIP

The Central Limit Theorem (6.3)
o Example – Graduate Record Examination (GRE) scores
 Suppose it is known that students’ scores on the GRE in the United States
have an average of 1600 and a standard deviation of 150. Suppose we
check 100 students’ scores who are enrolled at UCF and we calculate an
average score of 1525. If the students at UCF can be considered a random
sample of all students nationwide, how likely/unlikely is this? We must
first know something about the distribution of the sample mean, x .
o Properties of the Sampling Distribution of x
  x   (  population mean)
x 

(  population standard deviation) (standard error)
n
o Example – GRE scores (continued)
  x    1600

x 


150
 15
n
100
 We now know the mean and standard deviation of the sample mean.
However, we know nothing about the shape.
o Central Limit Theorem
 If a random sample of size n is drawn from any population, and n is
sufficiently large (n  30), then x will be approximately normally
distributed with mean  x and standard deviation  x .


NOTE: If a sample is drawn from a normal population, then x will be
normally distributed, regardless of the size of n.
o Example – GRE scores (continued)
 Since n = 100 which is at least 30, then we know the shape of x is
approximately normal with mean 1600 and standard deviation 15.
 What is the probability that 100 randomly selected students have an
average score of 1525 or lower on the GRE? We are trying to find
P( x 1500). Since x is approximately normal, we can convert this to
x  x
standard normal by using the formula z 
(substituting x for x in
x
our equation from Section 5.3). Using this formula yields
1525  1600
z
 5 , and the corresponding probability P(z<-5)  0.
15
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STA 2023
Chapter 6 – Sampling Distributions
o When will x be at least approximately normal?
Is n  30?
Normal
population?

Is x
normal?
Yes
Yes
No
No
Yes
No
Yes
No




Yes
Yes
Yes
No
 x   and

x 
n
Yes
Yes
Yes
Yes
o Example – Assume IQs are normally distributed with a mean of 100 and a
standard deviation of 15, and we randomly sample 25 people at a time.
 Will the distribution of the sample mean be normal? Yes, since the
original population is normal. Had the original population not been
normal then the distribution of the sample mean would not have been
normal.
 Sketch the distribution of x.
x
55


85
100
115
130
145
Sketch the distribution of the sample mean.
91

70
94
97
100
103
106
109
What is the probability that a randomly selected person has IQ above 109?
109  100
P(x > 109) = P(z >
) = P(z > .6) = .2743.
15
What is the probability that a random sample of 25 people has a mean IQ
109  100
above 109? P( x > 109) = P(z >
) = P(z > 3) = .0013.
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STA 2023
Chapter 6 – Sampling Distributions


What is the probability that a random sample of 100 people has a mean IQ
109  100
above 109? P( x > 109) = P(z >
) = P(z > 6)  0.
1 .5
What is the probability that a random sample of 4 people has a mean IQ
109  100
above 109? P( x > 109) = P(z >
) = P(z > 1.2) = .1151.
7 .5
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