Download Hypothesis Test

POLS 7000X
STATISTICS IN POLITICAL SCIENCE
CLASS 7
BROOKLYN COLLEGE-CUNY
SHANG E. HA
Leon-Guerrero and Frankfort-Nachmias,
Essentials of Statistics for a Diverse Society
Chapter 7: Testing Hypotheses (Class 7 & 8)
 Overview
 Assumptions of Statistical Hypothesis Testing
 Stating the Research and Null Hypotheses
 Determining What is Sufficiently Improbable: Probability
Values and Alpha
 Five Steps in Hypothesis Testing: A Summary
 Errors in Hypothesis Testing
 Testing Hypotheses About Two Samples
 The Sampling Distribution of the Difference Between
Means
 The t Statistic
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Hypothesis Test vs. Confidence Interval


Confidence Interval: to estimate the population
mean
Hypothesis Test: to evaluate whether a claim about
the population mean is valid
An Example: One Sample t-Test

An insurance company is reviewing its current policy.
When originally setting the premium rates they believed
that the average claim amount was $1,800. If the true
mean is actually higher than this, they could potentially
lose a lot of money. On the other hand, if the true mean
is actually lower than this, they could lower the premium
rates for greater competitiveness in the market. They
randomly select 100 claims, and calculate a sample mean
of $1,910 with a standard deviation of $500
For the above example…

We may construct a confidence interval to estimate
an average claim amount.
 95%

CI: 1,910 ± 1.96*(500/√100) = [1,812, 2,008]
We may conduct a hypothesis test to see if any
average claim amount is different from $1,800.
Null hypothesis vs. Alternative/Research
hypothesis

Null hypothesis; Ho: the default belief about the
population mean (always contains [=]!!!)


μ = $1,800
Alternative/Research hypothesis; Ha (or Hresearch): the
statement of a hypothesis that is opposite to the null
hypothesis



μ ≠ $1,800 (two-tailed test)
μ > $1,800 (one-tailed test)
μ < $1,800 (one-tailed test)
Hypothesis Test (Two-tailed test)



Under the presumption that Ho: μ=$1,800 is true, the
center of the sampling distribution of X is the value 1,800
The sample mean ($1,910) may be within the 95%
confidence interval or may be outside of it
It the sample mean is outside of the 95% confidence
interval, we are able to reject the null hypothesis
Hypothesis Test (two-tailed test)

Use t-score

|t| = |(1,910-1,800)/{(500/√100)}|= 2.2

As |t| increases, Ho is more likely to be rejected
Decision Rules I



We reject the null hypothesis if t-score is greater (in
absolute value) than the critical value
In our case, the critical value at 95% confidence level with
99 degrees of freedom is 1.984
The critical value is determined by the level of confidence
(α)and degrees of freedom (n-1)
Decision Rules II


p-value is the probability that the test statistics at
least as large in absolute value as the observed
value if Ho is true
We reject the null hypothesis if p-value is less than
α (significance level)
Things to Remember



Conclusions using (two-tailed) hypothesis tests are
consistent with conclusions using confidence intervals
We say “Do not reject Ho" rather than “Accept Ho"
It is better to report the p-value than to indicate merely
whether the result is “statistically significant" or not
Hypothesis Test (one-tailed test)


A different alternative hypothesis is sometimes used when a research
predicts a deviation from Ho in a particular direction
By law, an industrial plant can discharge no more than 500 gallons
of waste water per hour, on average, into a neighborhood lake. An
environmental action group believes this limit is being exceeded and
tests if the industrial plant indeed abides by the law. A random
sample of 90 hours over a period of a month reports a sample
mean of 525 gallons with a standard deviation of 110 gallon
One-tailed Test



In this example, we are only interested in if the industrial
plant discharge more than 500 gallons of waste water
per hour. Discharging less than 500 gallons per hour is
basically the same result as discharging exactly 500
gallons per hour
Null hypothesis (Ho): μ = 500
Alternative hypothesis (Ha): μ > 500
One-tailed test

The test statistics is again the t-score:
t


= (525-500)/(110/√90) = 2.16
The critical value at 95% confidence level with 89
degrees of freedom is 1.66
p-value = 0.017
Notes

For the same α, a one-tail test is easier to reject Ho than a two-tail test

In practice, two-tailed tests are more common than one-tailed tests

When deciding whether to use a one-tailed or a two-tailed test in a
particular exercise, consider the research question posed



Whether μ has changed vs. whether μ has increased
Is there a theoretical reason to predict that the change should be in a particular
direction?
You should not choose a one-tailed test simply because the sample mean is
greater or less than the test value.
Summary


What is your research question? Are the sample data obtained using
randomization? How large is n?
Based on your research question, explicitly state the null hypothesis and alternative
hypothesis. Is a two-tailed test more appropriate than a one-tailed test?

Decide the α-level (the probability of error you are willing the take)

Calculate the test statistics (t-score)


Compare the test statistics to the critical value for the given α (or compare p-value
to α)
Conclude and report your result