SPA 3e_ Teachers Edition _ Ch 6

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Lesson 6.3
The Sampling Distribution
of a Sample count
(The normal Approximation
to the Binomial)
L e A r n i n g T A r g e T S
d Calculate the mean and the standard deviation of the sampling distribution of
a sample count and interpret the standard deviation.
d Determine if the sampling distribution of a sample count is approximately
normal.
d If appropriate, use the normal approximation to the binomial distribution to
calculate probabilities involving a sample count.
In many cases, we are interested in the number of successes X in a random sample
from some population. For example, X 5 the number of defective flash drives in a
random sample of 10 flash drives or X 5 the number of Democrats in a random
sample of 1000 registered voters. To do probability calculations involving X, we want
an understanding of the sampling distribution of the sample count X.
DEFINITION Sampling distribution of the sample count X
The sampling distribution of the sample count X describes the distribution of values
taken by the sample count X in all possible samples of the same size from the same
population.
The sampling distribution of X is closely related to the binomial distributions that
you learned about in Lessons 5.3 and 5.4.
Suppose that a supplier inspects an SRS of 10 flash drives from a shipment of
10,000 flash drives in which 200 are defective. Let X 5 the number of bad flash drives
in the sample. This is not quite a binomial setting. Because we are sampling without
replacement, the independence condition is violated. The conditional probability that
the second flash drive chosen is bad changes when we know whether the first is good
or bad: P(second is bad | first is good) 5 200/9999 5 0.0200 but P(second is bad | first
is bad) 5 199/9999 5 0.0199. These probabilities are very close because removing
1 flash drive from a shipment of 10,000 changes the makeup of the remaining 9999
flash drives very little. The distribution of X is very close to the binomial distribution
with n 5 10 and p 5 0.02.
Answers continued
19. (a) Yes. Binary? “Success” 5 female is
selected. “Failure” 5 male is selected.
Independent? Knowing whether or not one
randomly selected student is a female tells you
nothing about whether or not another randomly
selected student is a female. Number? n 5 40.
Same probability? p = 55
104 = 0.529
(b) m X = np = 40(0.529) = 21.16
s X = "np(1 − p) = "40(0.529)(1 − 0.529)
= "40(0.529)(0.471) = "9.97 = 3.16
(c) P(X = 0) = 10 C 0 (0.529) 0 (1 − 0.529) 10
= 1(0.529) 0 (0.471) 10 = 0.0005
P(X = 1) = 10C 1 (0.529) 1 (1 − 0.529) 9
= 10(0.529) 1 (0.471) 9 = 0.006
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P(X ≤ 1) = P(X = 0) + P(X = 1)
= 0.0005 + 0.006 = 0.0065
If the professor were to randomly choose
students for the first 10 meetings, there is
less than a 1% chance that he would select
1 female or fewer purely by chance. Because
this is unlikely, we have convincing evidence
that his selection process is not really random.
Teaching Tip
To save time, Lessons 6.2 and/or 6.3
can be skipped without losing much
continuity in future chapters. However,
the other lessons in this chapter are
crucial to understanding much of the
remainder of the course.
Learning Target Key
The problems in the test bank are
keyed to the learning targets using
these numbers:
d 6.3.1
d 6.3.2
d 6.3.3
BELL RINGER
According to the manufacturer, the
true proportion of blue M&M’S® milk
chocolate candies is 0.24. If Mrs. Gallas
takes a random sample of 50 candies,
how many should she expect to be blue?
If she repeatedly takes random samples
of 50 candies, will she always get the
same number of blue candies? Why or
why not?
FYI
When sampling with replacement from
a finite population or sampling from an
infinite population, the distribution of
the sample count is exactly binomial.
However, these types of sampling
methods are not common in practice. It
is far more common to sample without
replacement from a finite population.
In this case, the sample count is usually
approximately binomial.
Lesson 6.3
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