Stat 104 – Lecture 25
Sampling Distributions
Qualitative/Categorical variable
Population Parameter: p known.
Population
Sample
Distribution of
Sample Proportion
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Simulation
• Population
– Reeses Pieces
statweb.calpoly.edu/chance/applets/Reeses/ReesesPieces.html
• Population Parameter
– Proportion of Orange Reeses Pieces
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Stat 104 – Lecture 25
Simulation
• Simple random sample of size
n=25.
• Repeat several times.
• Record the sample proportion of
orange Reeses Pieces.
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Distribution of sample proportion p̂
• Shape: Approximately Normal
• Center: The mean is p.
• Spread: The standard deviation is
p(1 − p )
n
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Stat 104 – Lecture 25
Conditions
• Must take a random sample
from the distribution.
• The sample size, n, must be
large enough, bigger than 20.
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Conditions
• 10% Condition
–When sampling without
replacement, the sample size
should be less than 10% of the
population size.
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Conditions
• Success/Failure Condition
– The sample size must be large enough
so that np and n(1- p) are both bigger
than 10.
– Note the text says bigger than 5 but
we are going to be cautious.
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Stat 104 – Lecture 25
Inference for Binomial p
Population Parameter?
Population
p
Inference
Sample
p̂
Sample Statistic
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Inference
• Propose a value for the population
proportion, p.
• Does the sample data support this
value?
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Example
• Are non-whites under-represented
on juries in Story County?
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Stat 104 – Lecture 25
Example
• According to the U.S. census,
Story County has 9.7% of its
population classified as non-white.
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Example
• Population: All people eligible for
jury duty in Story County.
• Parameter: Proportion of all people
eligible for jury who are nonwhite.
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Example
• For a random sample of 120 people
called for jury duty in Story
County only 3 are non-white. Is
this convincing evidence of underrepresentation of non-whites?
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Stat 104 – Lecture 25
Step 1- Set-Up
• State your null and alternative
hypotheses and define p.
– H0: p = 0.097
– HA: p < 0.097
–p is the proportion of non-whites
among all people in the jury pool for
Story County.
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Step 2
• Check conditions
–Random sampling condition
• Random sample taken
–Sample size
• 120 is bigger than 20
• 120 is less than 10% of the population
size.
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Step 2 (continued)
• Success/Failure condition
np0 = 120(0.097 ) = 11.64
n (1 − p0 ) = 120(0.903) = 108.36
both are greater than 10.
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Stat 104 – Lecture 25
Step 3 – Sample evidence
• Calculate the test statistic.
p̂ − p 0
0 . 025 − 0 . 097
=
p 0 (1 − p 0 )
0 . 097 ( 1 − 0 . 097 )
n
120
− 0 . 072
= − 2 . 67
z =
0 . 027
z =
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Step 4 – Probability value
• P-value is the probability of z
being less than –2.67.
• P-value = 0.0038
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Step 5 - Results
• Because the P-value is smaller than
0.05, we should reject the null
hypothesis.
• This is convincing evidence that
non-whites are under-represented in
the jury pool.
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