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Math 160 - Cooley
Intro to Statistics
OCC
Section 7.3 – The Sampling Distribution of the Sample Mean
Sampling Distribution of the Sample Mean for a Normally Distributed Variable
Suppose that a variable x of a population is normally distributed with mean μ and standard deviation σ. Then, for
samples of size n, the variable x is also normally distributed and has mean μ and standard deviation

.
n
The Central Limit Theorem (CLT)
For a relatively large sample size, the variable x is approximately normally distributed, regardless of the
distribution of the variable under consideration. The approximation becomes better with increasing sample size.
Sampling Distribution of the Sample Mean
Suppose that a variable x of a population has mean μ and standard deviation σ. Then, for samples of size n,
 The mean of variable x equals the population mean, or  x   ;
 The standard deviation of x equals the population standard deviation divided by the square root of the
sample size, or

.
n
 If x is normally distributed, so is x , regardless of sample size; and
 If the sample size is large, x is approximately normally distributed, regardless of the distribution of x.
Sampling Distributions of the sample mean for (a) normal, (b) reverse, and (c) uniform variables
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Math 160 - Cooley
Intro to Statistics
OCC
Section 7.3 – The Sampling Distribution of the Sample Mean
 Exercises:
1)
2)
A variable of a population has a mean of   35 and a standard deviation of   42 .
a)
If the variable is normally distributed, identify the sampling distribution of the sample mean for
samples of size 9.
b)
Can you answer part a) if the distribution of the variable under consideration is unknown?
Explain your answer.
c)
Can you answer part a) if the distribution of the variable under consideration is unknown, but
the sample size is 36 instead of 9? Why or why not?
Ciochetti et al. studied mortgage loans in the article “A Proportional Hazards Model of Commercial
Mortgage Default with Originator Bias”. According to the article, the loan amounts of loans originated
by a large insurance-company lender have a mean of $6.74 million with a standard deviation of $15.37
million. The variable “loan amount” is known to have a right-skewed distribution.
a)
Using units of millions of dollars, determine the sampling distribution of the sample mean for
samples of size 200. Interpret your result.
b)
Repeat part a) for samples of size 600.
c)
Why can you still answer parts a) and b) when the distribution of loan amounts is not normal,
but rather right skewed?
d)
What is the probability that the sampling error made in estimating the population mean loan
amount by the mean loan amount of a simple random sample of 200 loans will be at most $1
million?
 Sample Test Multiple Choice Questions:
3)
Let x represent the number which shows up when a balanced die is rolled. Then x is a random
variable with a uniform distribution. Let x denote the mean of the numbers obtained when the
die is rolled 3 times. Which of the following statements concerning the sampling distribution
of the mean is true?
A) x has a uniform distribution.
B) x is normally distributed.
C) x is approximately normally distributed. D) None of these.
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