Stat 104 – Lecture 22
Margin of Error
ME  z * SE ( pˆ )  z *
pˆ (1  pˆ )
n
The Margin of Error is the
furthest p̂ can be from p, for
a given confidence.
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Margin of Error
ME  z * SE ( pˆ )  z *
Confidence
z*
80%
90%
pˆ (1  pˆ )
n
95%
98%
99%
1.282 1.645 2 or 1.96 2.326 2.576
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Another Example
• Princeton Survey Research
Associates International
contacted 1,005 adults
nationwide October 20–21,
2010 and asked them the
following question:
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Stat 104 – Lecture 22
Another Example
• “California’s Prop 19 would
legalize marijuana in the
state. Would you support or
oppose a similar measure to
legalize marijuana in your
state?”
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Another Example
• 45% of the sample said that they
would support a measure to legalize
marijuana in their state.
• Construct a 98% confidence interval
for the population proportion who
would support a measure to legalize
marijuana in their state.
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Another Example
pˆ  0.45
pˆ (1  pˆ )
0.45(0.55)

 0.0157
n
1005
0.45  2.326(0.0157 ) to 0.45  2.326(0.0157 )
0.45  0.037 to 0.45  0.037
0.413 to 0.487
SE ( pˆ ) 
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2
Stat 104 – Lecture 22
Interpretation
• We are 98% confident that
between 41.3% and 48.7% of
all adults would support a
measure to legalize
marijuana in their state.
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Sample Size
• General formula for sample
size.
z *
n
pˆ (1  pˆ )
ME2
2
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Sample Size
• Priority example. ME = 0.01
with 98% confidence.
n
z *2 pˆ (1  pˆ )  2.3262 (0.45)(0.55)
ME2
0.012
n  13,390
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Stat 104 – Lecture 22
Inference: Hypothesis Test
• Propose a value for the
population proportion, p.
• Is the sample data consistent
with this value?
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Example
• A seed company claims that it’s new
corn hybrid has a 94% germination
rate.
• For a random sample of 500 seeds
only 452 germinate.
• What do you think of the company’s
claim?
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Example
• Population: All seeds for the
new corn hybrid.
• Parameter: Proportion of all
seeds for the new corn hybrid
that will germinate, p.
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Stat 104 – Lecture 22
Example
• Null Hypothesis
–H0: p = 0.94
• Alternative Hypothesis
–HA: p < 0.94
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Example
• How likely is it to get a sample
proportion as extreme as the
one we observe when taking a
random sample of 500 from a
population with p = 0.94?
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Example
• Sampling distribution of p̂
–Shape approximately normal.
–Mean: p = 0.94
–Standard Deviation:
0.94(0.06)
 0.0106
500
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Stat 104 – Lecture 22
Standardize
452
 0.904
500
0.904  0 .94  0.036
z

 3.40
0.0106
0.94 ( 0.06 )
500
pˆ 
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Use Table Z
z
.00
–3.4
– 3.3
– 3.2
.0003
.01
.02
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Interpretation
• Getting a sample proportion of
0.904 or less will happen only
0.03% (P-value = 0.0003) of the
time when taking random samples
of 500 from a population whose
population proportion is p = 0.94.
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