LESSON 15: DISTRIBUTION OF PROPORTION
Outline
• Sampling distribution of the proportion
– Context: attribute, proportion
– Expected value and standard deviation
– Continuity correction
– Cumulative probability
– Correction for small, finite population
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SAMPLING DISTRIBUTION OF THE PROPORTION
• Context: Suppose that each item can have two states. The
item either has an attribute or it does not. For example, an
item can be either defective or not.
• Usually, the states are good/bad, defective/non-defective,
yes/no etc.
• Sample proportion P is used to draw inference about the
unknown population proportion π
R
n
• Where R is the number of sample observations having a
particular attribute and n is the sample size
P=
2
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SAMPLING DISTRIBUTION OF THE PROPORTION
• Recall that binomial distribution applies in such cases.
However, since the sample size is usually large, binomial
distribution involves a large volume of computation.
• A more convenient approach is to use the normal
approximation to the binomial distribution. The expected
value and standard deviation are computed as follows:
E (P ) = π
π (1 − π )
n
• However, the binomial distribution of R (e.g., # of
defectives) is discrete and the normal distribution is
continuous. So, continuity correction may be required.
SD(P ) = σ P =
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SAMPLING DISTRIBUTION OF THE PROPORTION
• Continuity correction:
– Binomial probability P(R=r) is approximated by normal
probability P(r-0.5 ≤ R ≤ r+0.5)
– This simple rule gives rise to many adjustments
Binomial
Normal
P(R=r)
P(r-0.5 ≤ R≤ r+0.5)
P(a ≤ R ≤ b)
P(a-0.5 ≤ R ≤ b+0.5)
P(R ≤ r)
P(R ≤ r+0.5)
P(R ≥ r)
P(R ≥ r-0.5)
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SAMPLING DISTRIBUTION OF THE PROPORTION
• Summary: The cumulative probability is obtained as
follows:
E (P ) = π
SD(P ) = σ P =
π (1 − π )
n
 (r + 0.5 ) / n − π
r

Pr  P ≤  = Φ 
n
σP





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SAMPLING DISTRIBUTION OF THE PROPORTION
Example 1: A welding robot is judged to be operating
satisfactorily if it misses only 0.6% of its welds. A test is
performed involving 50 sample weds. If the number of
missed sample weds is more than 1, the robot will be
overhauled. Find the probability that a satisfactory robot will
be overhauled unnecessarily.
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SAMPLING DISTRIBUTION OF THE PROPORTION
• Correction for small population: The formula for standard
deviation requires a little correction if the population is
small. For small/finite population, the cumulative probability
is obtained as follows:
E (P ) = π
π (1 − π ) N − n
n
N −1
 (r + 0.5 ) / n − π 
r

Pr  P ≤  = Φ 

n
σP



SD(P ) = σ P =
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SAMPLING DISTRIBUTION OF THE PROPORTION
Example 2: A sample of 100 parts is to be tested from a
shipment of 400 items altogether. Although the inspector
does not know the proportion of defective in the shipment,
assume that this quantity is 0.02. Determine the probability
that P ≤ 0.05
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READING AND EXERCISES
Lesson 15
Reading:
Section 9-5, pp. 282-284
Exercises:
9-31, 9-32
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