Stat 104 – Lecture 27
Chapter 8
Quantitative variable
Population Parameters:
Population
μ
Sample
σ
known.
Sample
Mean
y
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Chapter 8
• Sampling from a population that
follows a Normal Model, μ , σ .
⎛⎜ σ ⎞⎟ to y + z * ⎛⎜ σ ⎞⎟
⎝ n⎠
⎝ n⎠
y−μ
z=
⎛⎜ σ ⎞⎟ , Table Z ⇒ P - value
⎝ n⎠
y − z*
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Chapter 9, Sections 1, 2
Quantitative variable
Population Parameters: unknown, σ
μ
Population
Sample
Sample
Mean,
y
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Stat 104 – Lecture 27
Unknown, σ
• If we do not know the value of the
population standard deviation we
cannot use the methods from
Chapter 8.
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Unknown, σ
• We can use the sample standard
deviation, s, as an estimate of
the population standard
deviation, σ .
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Unknown, σ
• We can NOT continue to use
the standard normal distribution
or Table Z.
• Why?
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Stat 104 – Lecture 27
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8
95% Confidence?
• Simulation illustrating repeating
the procedure.
• http://statweb.calpoly.edu/chance/
applets/ConfSim/ConfSim.html
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Stat 104 – Lecture 27
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Chapter 9, Sections 1, 2
• Confidence Interval for μ .
⎛ s ⎞
⎛ s ⎞
y − t* ⎜
⎟ to y + t * ⎜
⎟
⎝ n⎠
⎝ n⎠
• t* found in Table T, df = n – 1
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Chapter 9, Sections 1, 2
• Test statistic.
t=
y−μ
, Table T
s
n
⎛⎜ ⎞⎟
⎝ ⎠
⇒ P - value
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Stat 104 – Lecture 27
Example
• What is the mean alcohol
content of beer?
• A random sample of 10 beers is
taken and the alcohol content
(%) is measured.
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Example
• We do not know the value of the
population standard deviation,σ ,
for the population of all beers.
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Sample Data – Alcohol (%)
Molson
Canadian
5.19
Heineken
Dark
5.17
Michelob
Dark
4.76
O’Keefe
Canadian
4.96
Big Barrel
Lager
4.32
Olympia
Lager
4.78
Hamm’s
4.53
Miller Draft
4.85
Tsingtao
4.79
Guinness
Stout
4.27
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Stat 104 – Lecture 27
Sample Summary
• Sample size:
– n = 10
• Sample mean:
–
y = 4.762
• Sample standard deviation:
– s = 0.314
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Confidence Interval for μ
s ⎞
⎛ s ⎞
⎟ to y + t * ⎜
⎟
⎝ n⎠
⎝ n⎠
df = n − 1
⎛
y − t* ⎜
17
Inference for μ
• Do NOT use Table Z!
Table Z
• Use Table T instead!
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Stat 104 – Lecture 27
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95% Confidence Interval
•
y = 4.762
• n = 10
• s = 0.314
• t* = 2.262
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Calculation
⎛
y − t* ⎜
⎝
s ⎞
⎛ s ⎞
⎟ to y + t * ⎜
⎟
n⎠
⎝ n⎠
⎛ 0.314 ⎞
⎟
⎝ 10 ⎠
4.762 − 2.262⎜
⎛ 0.314 ⎞
⎟
⎝ 10 ⎠
to 4.762 + 2.262⎜
4.762 − 0.225 to 4.762 + 0.225
4.537 to 4.987
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