Transcript Lecture 38 - Bias and Variability

Bias and Variability
Lecture 38
Section 8.3
Wed, Mar 31, 2004
Unbiased Statistics
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A statistic is unbiased if its average
value equals the parameter that it is
estimating.
The variability of a statistic is a
measure of how spread out its
sampling distribution is.
All estimators exhibit some variability.
The Parameter
the parameter
Unbiased, Low Variability
the parameter
Unbiased, High Variability
the parameter
Biased, High Variability
the parameter
Biased, Low Variability
the parameter
Accuracy and Precision
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An unbiased statistic allows us to
make accurate estimates.
A low variability statistic allows us to
make precise estimates.
The best estimator is one that is
unbiased and with low variability.
Experiment
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Me: randBin(50, .1, 200)
Volunteer #1: randBin(50, .3, 200)
Volunteer #2: randBin(50, .5, 200)
Volunteer #3: randBin(50, .7, 200)
Volunteer #4: randBin(50, .9, 200)
It will take the TI-83 about 6 minutes.
Experiment
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Divide the list by 50 to get
proportions.
Store the results in list L1.
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Compute the statistics for L1.
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STO L1.
1-Var Stats L1.
What are the mean and standard
deviation?
The Sampling Distribution
of p^
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The distribution of p^ is approximately
normal.
The approximation is excellent if
np ≥ 5 and n(1 – p) ≥ 5.
The mean is p (the pop. proportion).
The standard deviation is
(p(1 – p)/n).
The Sampling Distribution
of p^
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Because n appears in the denominator
of the standard deviation,
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The standard deviation decreases as n
increases.
However, it does not decrease as fast
as n increases.
For example, if the sample is 100
times larger, the standard deviation is
only 10 times smaller.
Example
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A recent poll showed that 44% of the
voters favor George W. Bush in the
2004 election.
Suppose that the population
proportion p = 0.44.
Describe the distribution of p^ if the
sample size is n = 1000.
Example
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If we take a sample of 1000 voters,
what is the probability that at least
50% of them favor Bush?
The z-score of 0.50 is z = 3.82.
P(p^ > 0.50) = P(Z > 3.82)
= 0.0000667.
Example
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Find the probability that less than 42%
favor Bush.
Find the probability that between 42%
and 46% favor Bush.
Let’s Do It!
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Let’s do it! 8.5, p. 484 – Probabilities
about the Proportion of People with
Type B Blood.
Let’s do it! 8.6, p. 485 – Estimating
the Proportion of Patients with Side
Effects.
Let’s do it! 8.7, p. 487 – Testing
hypotheses about Smoking Habits.
Assignment
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Page 488: Exercises 1 – 15.