Stat 104 – Lecture 26
Alternatives
• H0 : p = po
– HA: p < p o, P-value = Pr < z
– HA: p > p o, P-value = Pr > z
– HA: p ≠ p o, P-value = Pr > |z|
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Alternatives
• H0 : p = po
• H A : p ≠ po
– If z is a negative value, P-value
=2(Probability less than z)
– If z is a positive value, P-value
=2(Probability greater than z)
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More on Testing
• 500 randomly selected U.S. adults
were asked the question: “Would
you be willing to pay much higher
taxes in order to protect the
environment?”
• 216 answered yes
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Stat 104 – Lecture 26
More on Testing
• Is this convincing evidence that the
proportion of all U.S. adults who
are willing to pay higher taxes is
different from 50%?
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Step 1 – Set-Up
• p is the proportion of all U.S.
adults who are willing to pay
higher taxes in order to protect the
environment
– H0: p = 0.50
– HA: p ≠ 0.50
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Step 2
• Check conditions
–Random sampling condition
–10% condition
–Success/Failure condition
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Stat 104 – Lecture 26
Step 3 – Sample Evidence
• Calculate the test statistic.
p̂ − p 0
0 . 432 − 0 .5
=
0 . 5 (0 . 5 )
p 0 (1 − p 0 )
500
n
− 0 . 068
= − 3 . 04
z=
0 . 0224
z=
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Step 4 – Probability value
• Calculate the P-value.
• The probability of z being less than
–3.04 is 0.0012.
• The P-value is 2(0.0012) = 0.0024
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Step 5 – Results
• Reject the null hypothesis because
the P-value is less than 0.05.
• There is convincing evidence that
the proportion of the U.S. adult
population willing to pay more
taxes to protect the environment is
different from 50%.
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Stat 104 – Lecture 26
Example
• Population: Registered voters in
the U.S.
• Parameter: Proportion of U.S.
voters who believe global warming
exists. Unknown!
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Example
• Sample: 900 randomly selected
registered voters nationwide. FOX
News/Opinion Dynamics Poll, Jan.
30-31, 2007.
• Statistic: 737 of the 900 voters in
the sample (82%) believe global
warming exists.
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Approximately Normal
• 95% of the time the sample
proportion, p̂ , will be between
p − 1.96
p( 1 − p )
p( 1 − p )
and p + 1.96
n
n
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Stat 104 – Lecture 26
Standard Deviation
• Because p, the population proportion
is not known, the standard deviation
SD( pˆ ) =
p(1 − p)
n
is also unknown.
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Standard Error
• Substitute p̂ as our estimate (best
guess) of p.
• The standard error of p̂ is:
SE ( pˆ ) =
pˆ (1 − pˆ )
n
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Confidence Interval for p
• We are 95% confident that p will
fall between
p̂ − 1.96
p̂( 1 − p̂ )
p̂( 1 − p̂ )
and p̂ + 1.96
n
n
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Stat 104 – Lecture 26
Example
p̂ = 0.82
p̂(1 − p̂ )
0.82(0.18)
=
= 0.013
n
900
0.82 − 1.96(0.013) to 0.82 + 1.96(0.013)
0.82 − 0.025 to 0.82 + 0.025
0.795 to 0.845
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Interpretation
• We are 95% confident that the
population proportion of all
registered voters in the U.S. who
believe global warming exists is
between 79.5% and 84.5%.
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Interpretation
• Plausible values for the population
parameter p.
• 95% confidence in the process that
produced this interval.
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Stat 104 – Lecture 26
95% Confidence
• If one were to repeatedly sample at
random 900 registered voters and
compute a 95% confidence interval
for each sample, 95% of the
intervals produced would contain
the population proportion p.
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Simulation
http://statweb.calpoly.edu/chance/ap
plets/Confsim/Confsim.html
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