Economics 105: Statistics
• Go over GH 11 & 12
• GH 13 & 14 due Thursday
Hypothesis Testing for  Using t
• Pharmaceutical manufacturer is concerned about impurity concentration in pills,
not wanting it to be different than 3%. A random sample of 16 pills was drawn
and found to have a mean impurity level of 3.07% and a standard deviation (s) of
.6%.
• Test the following hypothesis at the 1% level on the test statistic scale.
H0 : m = 3
H1 : m ¹ 3
• Perform the test on the sample statistic scale.
• What is the p-value for this test?
• Calculate the 99% confidence interval.
When to use z or t-test for H0: = 0
Xi~N

n ≥ 30
Xi not ~ N
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s
n<30
n ≥ 30
n<30
n ≥ 30
s
n<30
n ≥ 30
n<30
Nonparametric versus Parametric
Hypothesis Testing
Parametric Tests
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require the estimation of one or more unknown
parameters (e.g., population mean or variance).
often, unrealistic assumptions are made about the
normality of the underlying population.
large sample sizes are often required to invoke the
Central Limit Theorem
typically more powerful if normality can be
assumed
Nonparametric versus Parametric
Hypothesis Testing
Nonparametric Tests (“distribution-free”)
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usually focus on the sign or rank of the data rather
than the exact numerical value
do not specify the shape of the parent population
can often be used in smaller samples
can be used for ordinal data
usually more powerful if normality can’t be assumed
require special tables of critical values if small n
Nonparametric Counterparts
Source: Doane and Seward (2009), Applied Statistics in Business & Economics, 2e; McGraw-Hill
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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The one-sample runs test (Wald-Wolfowitz test) detects
non-randomness.
Ask – Is each observation in a sequence of binary
events independent of its predecessor?
A nonrandom pattern suggests that the observations are
not independent.
The hypotheses are
H0: Events follow a random pattern
H1: Events do not follow a random pattern
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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To test the hypothesis, first count the number of
outcomes of each type.
n1 = number of outcomes of the first type
n2 = number of outcomes of the second type
n = total sample size = n1 + n2
A run is a series of consecutive outcomes of the same
type, surrounded by a sequence of outcomes of the other
type.
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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For example, consider the following series representing
44 defective (D) or acceptable (A) computer chips:
DAAAAAAADDDDAAAAAAAADDAAAAAAAADDDDAAA
AAAAAAA
• The grouped sequences are:
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A run can be a single outcome if it is preceded and
followed by outcomes of the other type.
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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There are 8 runs (R = 8).
n1 = number of defective chips (D) = 11
n2 = number of acceptable chips (A) = 33
n = total sample size = n1 + n2 = 11 + 33 = 44
The hypotheses are:
H0: Defects follow a random sequence
H1: Defects follow a nonrandom sequence
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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When n1 > 10 and n2 > 10, then the number of runs R may be
assumed ~ N(m ,s 2 )
R
calc
R
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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Decision rule for large-sample runs tests at .01 level
• Critical values on test statistic scale = +/- 2.576
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Test statistic mR = 17.5, s R = 2.438
8 -17.5
Z=
= -3.90
2.438
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Conclusion?
Nonparametric Counterparts
Source: Doane and Seward (2009), Applied Statistics in Business & Economics, 2e; McGraw-Hill
Wilcoxon Signed-Rank Test
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Compares a single sample median with a benchmark
using only ranks of the data instead of the original
observations.
Can also be used to compare paired observations
• That’s a two-sample test
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Advantages are
- freedom from the normality assumption
- robustness to outliers
- applicability to ordinal data
The population should be roughly symmetric
Wilcoxon Signed-Rank Test
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To compare the sample median (M) with a
benchmark median (M0), the hypotheses are:
Wilcoxon Signed-Rank Test
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Calculate the difference between each observation and the
hypothesized median.
Rank the differences from smallest to largest by absolute
value. Same rank only if same sign before abs value.
Add the ranks of the positive differences to obtain the rank
sum W.
Wilcoxon Signed-Rank Test
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For small samples, a special table is required to
obtain critical values.
For large samples (n > 20), the test statistic is
approximately normal.
H 0 : M specialty = 20.2
H1 : M specialty ¹ 20.2
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Use Excel to get a p-value
Reject H0 if p-value < a
Wilcoxon Signed-Rank Test
pvalue = 2*(1- normsdist(.4344)) = .6639
Hypothesis Testing for  Using z
• A marketing company claims that it receives an
8% response rate from its mailings to potential
customers. To test this claim, a random sample of
500 potential customers were surveyed. 25
responded.
• a =.05
• Calculate power and graph a “power curve”
• Reminder: CI for  uses p in standard error, not !
– because CI does not assume H0 is true