** Are You Prepared to Learn Today?
** Did You Write Your Questions on the Board?
I. What is Chap. 18 about?
Sampling Distribution Models
II. Why should we learn them?
These models allow us to see different random samples
will give us different sample statistics. The good
news is: we can describe their distribution with
certain statistical models and use these models to
make confident conclusions about the population
parameter of interest based on our sample statistics.
III. How do we learn them?
Learn the Key Points:
1. Categorical Data and the Sample Proportions of a
category of interest:
 We want to find the distribution of the sample
proportion of students working on campus when
we survey any random 50 SSU students. Suppose
the true population parameter, p, is 30% (30%
of all 8000 SSU students are working on
campus).
What is the random variable? ____
Is it categorical or quantitative? ___
Can you simulate a survey of 50 random SSU
students and calculate the proportion of them
working on campus? ______
 Check the conditions:
Ai-Chu Wu, Ph. D.
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1). Random Sample:
The 50 students are from a simple random sample.
2). 10% Condition: The sample is less than 10% of
the population, n<10%N.
Since 50 is less than 10% of all SSU students, we
assume the 50 students are independent draws from
the population.
3). Success/Failure Condition: The sample size is
large enough; the expected hits and misses are at
least 10.
Since np = 50(0.30) = 15 and nq = 50(0.70) = 35
are both >= 10. We don’t need to worry about
skewed distribution.
 Name the model: When the above three conditions
are met, we can describe the sampling
distribution with this model: Normal (p,
(pq/n)), here q=1-p
The sampling model for the proportion of 50
random SSU students that work on campus has a
mean of ___ and a standard deviation of _____.
In mathematical shorthand, we say the model for
__ is N(__, __).
 Make confident conclusions: According to the
Normal model we learned in Chapter 6, we expect
68% of all random samples of 50 SSU students to
have proportions of on-campus workers between
___ and ____, 95% of the samples to have
proportions between ____ and ____, and 99.7% of
the samples to have proportions between ____
and ____.
2. Quantitative Data and Sample Means
 We want to find the distribution of the average
course-load (in units) of any 50 random SSU
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students.
Let us suppose the true mean is 15.
 Check the conditions:
1). Random Sample:
The 50 students are from a simple random sample.
2). 10% Condition: The sample is less than 10% of
the population, n<10%N.
Since 50 is less than 10% of all SSU students, we
assume the 50 students are independent draws from
the population.
3). The sample size is large enough, so we don’t
need to worry about skewed distribution.
 Name the model:
 Make confident conclusions:
3. Central Limit Theorem: The foundation for the
two models above.
 When the data are normal:
 When the data are not normal:
IV. Check what we have learned:
 Practice Quiz
1. It is generally believed that nearsightedness
affects about 12% of children. A school
district gives vision tests to 133 incoming
kindergarten children.
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a. Describe the sampling distribution model for
the sample proportion by naming the
model and telling its mean and standard
deviation. Justify your answer.
b. Sketch and clearly label the model.
c. What is the probability that in this group
over 15% of the children will be found to be
nearsighted?
2. The average composite ACT score for Ohio
students who took the test in 2003 was 21.4.
Assume that the standard deviation is 1.05. In
a random sample of 25 students who took the
exam in 2003, what is the probability that the
average composite ACT score is 22 or more?
(Make sure to identify the sampling
distribution you use and check all necessary
conditions.)
 Textbook HW
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