ST361: Ch 5.5 + Ch 5.6 Sampling Distribution
Topics:
I. What is Sampling Distribution?
II. Sampling Distribution of a Sample Mean X
(a) X ~ Normal Distribution
(b) X ~ Non-normal Distribution
III. Central Limit Theorem
IV. Sampling Distribution of the Sample Proportion p
---------------------------------------------------------------------------------------------------------------------------I. Sampling Distribution

Population vs. Sample:

A Parameter is_____________________________________________

A Statistic is _________________________________________________

The observed value of statistic depends on the particular sample; hence it ________
from sample to sample. Such variability is called ____________________________

The probability distribution of the statistics is called _________________________
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Ex1. A neighborhood has 5 houses A, B, C, D and E. They respectively have 3, 2, 5, 3, and 4
bedrooms. We randomly draw 3 houses at a time and calculate the sample statistics
median and mean. What is the sampling distribution of the sample median? What is the
sampling distribution of the sample mean?

Population =

Variable of interest =

Sample =
Houses drawn in the
sample
# of bedrooms
Sample median
Sample mean
ABC
3,2,5
3
10/3=3.3
ABE
3,2,4
3
9/3 = 3
ACD
3,5,3
3
11/3 = 3.7
ACE
3,5,4
ADE
3,3,4
ABD
10/3 = 3.3
BCD
BCE
2,5,4
4
11/3
BDE
2,3,4
3
9/3 = 3
CDE
5,3,4
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II. Sampling Distribution of a Sample Mean X
Let X be the sample mean of a random sample X1 , X 2 ,..., X n from a population mean  and
SD  . (That is, X 
X1  X1    X n
.) We want to know the sampling distribution of X .
n
 If X ~ Normal (mean=  , SD=  ). Then X , the mean of a random sample of n
observations

follows a _____________________, with mean

X =

X
______________, and
X
=
X
and standard deviation
X .
____________
is also called standard error (SE) of X , or Standard error of the mean
Ex 2. Thousands of boxes contain nuts. The weights are normally distributed with mean  =1 lb
and SD  =0.01 lb. We inspect 4 boxes and get their weights X 1 , X 2 , X 3 , X 4 . The sample
mean is X 
X1  X 2  X 3  X 4
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(a) What is the sampling distribution of X ? Mean and SE of X ?
(b) What is the probability that X lies between 0.99 and 1.01 lb?
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 X ~ any non-normal distribution with mean=  , SD=  . The sampling distribution of X
based on samples of size n is
(a) If n is small (i.e., _______________ ), then

Distribution:

Mean  X and SE  X :
(b) If n is large (i.e., ________________ ), then

Distribution:

Mean  X and SE  X :
 These results follow from Central Limit Theorem (CLT)
III. Central Limit Theorem
Assume X follows an arbitrary distribution with mean  and SD  .
When sample size is sufficiently large (i.e., n  30), the sample distribution of X always
follows normal distribution with mean  and SE


n
Usually the ____________________ a distribution is, the __________ the sample
size will need to ensure normality of X
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Ex3. Let X be the number of major defects for each new automobile tested. Suppose the number
of such defects for a certain model is with mean  =3.2 and SD  =2.4. A sample of 100
new cars is collected.
(a) What is the sampling distribution of X based on samples of size 100? What is its center
and what is the SE of X ?
(b) What is the probability that the sample average number of major defects exceeds 4?
 Comments:

If X be the sample mean of a random sample X1 , X 2 ,..., X n from a population mean
 and SD  , then regardless of the sample size n and the distribution of X,
 X  ____________,  X  ___________________

The variation of sample means is ________( , ,or  ) than variation of the original data
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
As sample size n increases,  x (the SE of X ) ____________, and the shape of the
sampling distribution becomes __________________ . This implies higher probability
around its mean  .
Ex4. The heights of college age students (denoted by X) are known to have mean  =115 and
SD  =30.
(a) What is the sampling distribution of X , the average height of 36 college age students?
What are the mean and SE of the sampling distribution of X ?
(b) What is the sampling distribution of X based on samples of 9 college age students? What
are the mean and SE of the sampling distribution of X ?
(c) Assume that we were told that the heights of college age students are normally
distributed. What is the sampling distribution of X based on samples of 9 college age
students? What are the mean and SE?
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IV. Sampling Distribution of a Sample Proportion p
Ex. Consider a basket containing 100 balls with 2 colors: Red and White. The proportion of
Red balls is denoted by  (and is not known). Assume 20 balls were randomly picked
from the basket with replacement, and 14 balls out of the 20 balls were red.
(1) In the sample, what is the proportion of red balls?
(2) We refer such quantity, 14 / 20, as _________________________and denote it by_____.
(Note that ____________________________________________) Our question of
interests: what is the distribution of the sample proportion p ?
Thoughts: we can think a r.v such that X = 1 if “red” and X=0 if “not red”.
Then p can be view as ________________________________.
That is, p is ______________________.
Thus by ___________________________, p ~______________if n large.
(However, different criteria for “large n ” are needed here.)
Sampling Distribution of p
(a) If _large n (i.e.,____________________________), then the sample proportion p has

A ________________________ ( by ___________________________ )

Mean (denoted by  p )   , and SE (denoted by  p ) 
 1   
n
(b) If _small n (i.e.,_____________________________), then the sample proportion p has

________________________________

Mean (denoted by  p )  ______, and SE (denoted by  p ) = ____________
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Ex5. In the population, the proportion of defectives  =12%.
(a) What is the sampling distribution of p based on 100 observations? What is the mean?
What is the standard error?
(b) What is the probability that p <0.05?
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