1. Stat310 Confidence intervals
Hadley Wickham
Thursday, 15 April 2010
2. Quiz
• Pick up quiz on your way in
• Start at 1pm
• Finish at 1:10pm
• Closed book
Thursday, 15 April 2010
3. 1. Test extra credit
2. Inference roadmap
3. Steps for making a confidence
interval
4. One more sampling distribution (the
t-distribution)
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4. I s rt
til y p
pa
Test makeup
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Homework graded out 10
4% of overall grade (= 20% of two tests)
Will act as extra credit for the test. (i.e.
there is no penalty you don’t do it)
Due next Thursday.
The extra 5% of the grade will be
distributed across all other assessment.
Thursday, 15 April 2010
5. What we want to do
Given data:
• Estimate true value of parameter
(last week)
• Quantify uncertainty of estimate
(today)
• Test whether true value is a certain value
(Thursday)
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6. Tools
• Construct an estimator
• Method of moments
• Maximum likelihood
• Work out its distribution
• Sampling distribution of mean
• Sampling distribution of variance
• General properties of ML (not in this course)
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7. Set up
I repeated an experiment defined by Poisson(λ)
10 times, and recorded the following results: 6
11 10 6 12 7 8 5 7 10
The MLE of λ is 8.2, and its standard deviation
is 0.90.
What is the distribution of the estimate?
(Remember that it’s a mean) Can you construct
an interval that will contain λ 95% of the time?
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8. Steps
1. Identify distribution that connects estimator
and true value (4 choices).
2. Form confidence interval for known
(sampling) distribution, and work out bounds.
3. Back transform.
4. Write as interval.
5. Plug in sample estimates (actual numbers).
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9. Your turn
Work through the steps on the handout.
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10. Confidence interval
A confidence interval is a simple numerical
summary of the uncertainty of an estimate.
A 95% confidence interval will contain the
true value 95% of the time.
An additional constraint is that we want
the confidence interval to be a short as
possible.
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11. Each line = 95% confidence
interval from one experiment
12
11
10
9
8
50 100 150 200
expt
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12. Horizontal line = true value
12
11
10
9
8
50 100 150 200
expt
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13. Red intervals don’t include true
value
12
11
10
9
8
50 100 150 200
expt
There are 13 red lines and 200
experiments. Is this an ok
interval?
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14. Your turn
What’s wrong with a statement like this:
P(2 < μ < 6) = 0.95
?
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15. Steps
Identify distribution that connects
estimator and true value.
Form confidence interval for
known (sampling) distribution.
Write as probability statement.
Back transform.
Write as interval.
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16. Xi iid, and n large:
¯n − µ .
X
√ ∼Z
σ/ n
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17. iid 2
Xi ∼ Normal(µ, σ )
(n − 1)S2
2
∼ χ (n − 1)
2
σ
X ¯n − µ
√ ∼Z
σ/ n
X ¯n − µ
√ ∼ tn−1
s/ n
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18. 0.3
df
1
dens
0.2 2
15
Inf
0.1
−3 −2 −1 0 1 2 3
x
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19. Properties of the t-dist
Heavier tails compared to the normal
distribution.
lim tn = Z
n→∞
Practically, if n > 30, the t distribution is
practically equivalent to the normal.
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20. t-tables
Basically the same as the standard
normal. But one table for each value of
degrees of freedom.
Easiest to use calculator or computer:
http://www.stat.tamu.edu/~west/applets/
tdemo.html
(For homework, use this applet, for final, I’ll give
you a small table, if necessary)
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22. Your turn
We perform the experiment an experiment to
measure the speed of sound and repeat it 10
times: 340 333 334 332 333 336 350 348 331
344 (mean: 338, sd: 7.01)
Assuming Xi ~ Normal(μ, σ2), what is an
estimate of the speed of sound? What is the
error (sd) of this estimate? Give an interval
that we’re 95% certain the true speed of
sound lies in.
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23. Example
340 333 334 332 333 336 350 348 331
344 (mean: 338, sd: 7.01)
If not known: (333, 342) (2.23)
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24. Steps
Identify distribution that connects
estimator and true value.
Form confidence interval for
known (sampling) distribution.
Write as probability statement.
Back transform.
Write as interval.
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25. Steps
Want P(a < Q < b) = 1 - α, and b - a to be
as small as possible.
If Q is symmetric, P(-a < Q < a) = 1 - α. So
a = F(α/2), and there is no interval smaller.
If Q isn’t symmetric, pick a = F(α/2),
b = F(1 - α/2), but there might be a shorter
interval.
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26. Example
We want a 90% confidence interval, then
two possible ends for the interval are
F(0.05) and F(0.95)
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27. Reading
Read the rest of chapter 6.
Everything else is just examples of the
general method we learned today.
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