Transcript Analytical Methods I

Chapter 9: Introduction to
Inference
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Thumbtack Activity
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Toss your thumbtack in the air and record
whether it lands either point up (U) or point
down (D). Do this 25 times (n=25).
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Calculate p-hat.
Repeat the above process two more times,
for a total of three estimates.
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Record your p-hat on a separate post-it note.
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We’ve just begun a sampling
distribution.
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Strictly speaking, a sampling distribution is:
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A theoretical distribution of the values of a statistic
(in our case, the proportion) in all possible
samples of the same size (n=25 here) from the
same population.
Sampling Variability:
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The value of a statistic varies from sample-tosample in repeated random sampling.
We do not expect to get the same exact value for
the statistic for each sample!
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Definitions
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Parameter:
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A number that describes the population of interest.
Rarely do we know its value, because we do not (typically)
have all values of all individuals from a population.
We use µ and σ for the mean and standard deviation of a
population.
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P and σp for proportions.
Statistic:
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A number that describes a sample. We often use a statistic
to estimate an unknown parameter.
We use x-bar and s for the mean and standard deviation of
a sample.
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P-hat and σp-hat for proportions.
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Sampling Distribution
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The sampling distribution answers the
question, “What would happen if we
repeated the sample or experiment
many times?”
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Formal statistical inference is based on the
sampling distribution of statistics.
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Inference
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Inference is the statistical process by
which we use information collected
from a sample to infer something about
the population of interest.
Two main types of inference:
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Interval estimation (Section 9.1)
Tests of significance (Section 9.2)
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Constructing Confidence Intervals
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Back to the thumbtack activity …
Interpretation of 95% C.I.:
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If the sampling distribution is approximately
normal, then the 68-95-99.7 rule tells us that
about 95% of all p-hat values will be within two
standard deviations of p (upon repeated
samplings). If p-hat is within two standard
deviations of p, then p is within two standard
deviations of p-hat. So about 95% of the time, the
confidence interval will contain the true population
parameter p.
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Internet Demonstration, C.I.
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http://bcs.whfreeman.com/yates/pages/bcsmain.asp?s=00020&n=99000&i=99020.01&v=category&o=&ns
=0&uid=0&rau=0
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Interpretation of 95% CI (Commit to memory!)
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95% of all confidence intervals
constructed in the same manner will
contain the true population parameter.
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5% of the time they will not.
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p. 492
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Finding a 95% C.I.
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Practice
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See example 9.3, p. 495
Exercises 9.1-9.4, p. 495
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Creating the C.I.
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Estimate +/- Margin of error
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Another practice problem
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9.5, p. 496
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p. 496
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Finding a confidence interval, general form
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Figure 9.5, p. 502
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Practice
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9.9 and 9.10, p. 505
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Confidence intervals with the calculator
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9.2 Significance Testing
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An evolutionary psychologist at Harvard University
claims that 80% (p=0.80) of American adults
believes in the theory of evolution. To test his claim,
he takes an SRS of 1,120 adults. Here are the
results:
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851 said “Yes” when asked, “Do you believe in the theory of
evolution?”
What is the proportion who said yes?
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Is this enough evidence to say that the proportion of adults
who do not believe in the theory of evolution is different
from 0.80?
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Example, cont.
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This requires a significance test:
Hypotheses:
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Ho: p=0.80
Ha: p≠0.80
Let’s use our calculators to conduct the
appropriate test:
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5: 1-prop ztest
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Example Results
P-value
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p. 516
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Hypotheses
Alternate hypothesis Ha:
Can be one-sided (Ha: p> some number or p< some number)
or two-sided (Ha: p≠ some number)
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HW
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9.24-9.26, p. 521
Reading: pp. 509-525
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p. 519
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Sampling Applet
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http://www.ruf.rice.edu/~lane/stat_sim/
sampling_dist/
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