Download Administration

5/21/15
Administration
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Seating chart (name and email address)
Coffee sign up (first name)
New syllabus posted (one paper dropped)
Exam will be open book and notes
Questions?
•  Back to sampling methods….
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Sampling Methods
•  Goal: sample is representative of the
population
•  Ideal: random sample
–  every possible sample equally likely to be
drawn
•  Check: compare sample characteristics to
known population parameters (if available)
Examples (p. 30)
•  Are the samples random?
–  Letters from constituents
•  Goal: determine whether constituents favor bill
•  Sample: views expressed in letters
–  Survey of law students
•  Goal: measure importance of opportunity to travel to future
choice of employment
•  Sample: survey of law students taking international trade
•  Powerful critiques include arguments about how
the sample might be biased (results produced
using sample can’t be used to draw conclusions
about the population of interest)
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Hypothesis Testing
•  Start with hypothesis derived from some
claim or theory
–  The average LSAT score for all U.S. law
students in 2007 was 151.
•  Specify null and alternative hypotheses
–  Null hypothesis: Average LSAT = 151
–  Alternative hypothesis:
•  Two-sided test: Average LSAT ≠ 151
•  One-sided test: Average LSAT > 151 (or average
LSAT < 151)
Hypothesis testing: how to do it
1. Formulate hypotheses
2. Draw sample
3. Calculate sample mean
4. Calculate (sample)
standard deviation
If sample size is sufficiently
large, the sample mean will
approximate the population
mean.
z-statistic
(or t-statistic)
P-value
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Example of a Test Statistic
•  z statistic
–  z = (sample mean – hypothesized population mean) /
(sample std dev / square root of the sample size)
•  Test statistics have known distributions!
Central Limit Theorem
•  The sampling distribution for means of
samples consisting of 30 values or more
will be roughly normal!!
•  This is true no matter what the initial
distribution looks like!!!
•  EXAMPLES….
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The Normal Distribution
68% of observations fall within 1 standard deviation of the mean
(RED)
95% fall within 2 standard deviations from the mean
(RED and GREEN)
99.7% fall within 3 standard deviations from the mean
(ALL COLORS)
P-value
•  The p-value is the probability of observing
a test statistic at least as extreme as the
observed test statistic (e.g., z or t) given
that the null hypothesis is true.
•  We know the probability because we know
the distribution of the test statistic (it’s
normal!)
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Interpretation of Results
Reminder: The p-value is the probability
of observing a test statistic at least as
extreme as the observed test statistic
(e.g., z or t) given that the null hypothesis
is true (e.g., LSAT mean = 151)
Example 1
Example 2
Sample mean = 155
t = 5.5
p = 0.002
Sample mean = 152
t = 1.1
p = 0.20
Therefore, reject null in
favor of alternative
Therefore, evidence
insufficient to reject null
Data do not support the
guess (“prior”)
Data support guess
Relationship between p-values and test statistics
Two-sided
t-statistic
p-value
Reject null at the…
| t | > 2.58 *
p < 0.01
1% level (high level of
confidence in rejection)
2.58 > | t | > 1.96
0.01 < p < 0.05
5% level (medium level
of confidence in
rejection)
1.96 > | t | > 1.65
0.05 < p < 0.10
10% level (low level of
confidence in rejection)
| t | < 1.65
p > 0.10
Cannot reject null
* <----------------|--------------------|--------------------|-------------->
- 2.58
0
2.58
(guess)
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Type I and Type II Errors
Null: Average population LSAT score is 151.
Alternative: Average population LSAT score is not equal to 151.
NULL HYPOTHESIS
TRUE
ACCEPT
NULL
Life is good
REJECT
NULL
Type I error
(“significance”)
RESULT
OF TEST
FALSE
Type II error
(“power”)
Life is good
α
Example: Test a coin for fairness
1. Identify a testable hypothesis
2. Toss the coin a number of times (gather
data)
3. Analyze the results
4. Draw an inference about the fairness of the
coin based on the sample
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Example
Null hypothesis: p(head) = 50%
Alternative hypothesis: p(head) ≠ 50%
Data: Of 1,000 tosses, 540 came up heads.
Example
z-value = 2.53
(test statistic)
p-value = 0.01
If the null hypothesis is true (i.e., the coin is fair), the
probability that we would observe a z-value this
extreme or more extreme is only 1%!!
Inference: At a 95% confidence level, we reject
the null hypothesis in favor of the alternative
(i.e., the data do not allow us to conclude that
the coin is fair).
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