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March 14th, by 5pm.
Political Science 15
Lecture 13:
Hypothesis Testing, Interpretation of
Hypothesis Tests
Hypothesis Testing
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If the null hypothesis is true, our sample statistic will
come from a normal distribution centered on that
number. If it is “too far” away (furthest 5% of the
distribution from the center), we reject the null
hypothesis as likely to be false.
Calculating a Test Statistic
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How do we know if our sample statistic falls
inside or outside the critical values for our
hypothesis test?
We must calculate a test statistic. In this case, the
number of standard deviations our sample
statistic is from the null hypothesis.
If we know the standard deviation of the
sampling distribution, we can calculate a z score:
Estimating Population Variance
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Using z scores to test our hypotheses relies on the
assumption that we know the standard deviation of the
variable we are testing. In reality we will not know this.
We instead estimate the standard deviation with the
square root of S2:
We then use a slightly different distribution to account
for the fact we had to estimate . This is the t
distribution.
The t Distribution
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The t distribution is similar to the normal, but
more spread out to account for the additional
uncertainty that comes from estimating . This
will make our critical values slightly larger.
We say the t distribution is distributed with n-1
degrees of freedom, where n is sample size. More
degrees of freedom mean more information was
used to determine our distribution.
As sample size increases, the t approximates the
normal.
The t Distribution
k = degrees of freedom
Hypothesis Test with a t
Example #1
We hypothesize the mean level of education in the US
is 14 years. H0 = 14. HA 14.
 We calculate the mean level of education in our sample.
That mean comes out to 15.
 We estimate S in our sample to be 2. We have 30
observations.
 Our test statistic is a t score.
t = (15 – 14)/(2 / 30) = 2.74.
 With a level of significance = 5%, our critical values in
a t distribution with 29 d.f. are  2.045.
 Our test statistic falls outside this range. Thus, we
reject the null hypothesis.
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Hypothesis Test with a t
Example #2
We hypothesize that IMF loans cause more political
instability. H0 for our regression slope = 0. HA 0.
 We calculate a regression line. The slope coefficient on
IMF loans is 2. We calculate S = 6.
 Our test statistic is a t score.
t = (2 – 0)/(6 / 150) = 4.08.
 With a level of significance = 5%, our critical values in
a t distribution with 149 d.f. are  1.98.
 Our test statistic falls outside this range. Thus, we
reject the null hypothesis. IMF loans do seem to have
a positive effect on political instability.
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Hypothesis Testing with H0=0
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If we reject H0=0, we say that variable is
statistically significant -- that is, we can reject the
null hypothesis that it has no effect.
This is not the same as substantively significant.
Something can be statistically significant but
have a tiny effect on the dependent variable.
Most statistical programs (including SPSS) will
automatically perform a t test on each
coefficient in the regression, using 0 as the null
hypothesis.
Hypothesis Testing in Practice
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We’ve now seen the basics of hypothesis testing
-- setting up null and alternative hypotheses,
estimating a test statistic, and determining
whether to reject or fail to reject the null
hypothesis based on this test statistic.
Now we will see how hypothesis tests are
actually used in the social sciences. We will
focus mostly on regression.
Hypothesis Testing in Regression
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In most cases we are testing whether a
relationship is positive or negative, so we test
the coefficients in a regression with H0= 0.
Most statistical programs (including SPSS) will
automatically perform a t test on each
coefficient in the regression, using 0 as the null
hypothesis.
If we reject H0= 0 for the coefficient on a
variable we say that variable is statistically
significant -- that is, we can reject the hypothesis
that it has no effect.
Standard Errors
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The standard error of a sample statistic is just
our estimate of the standard deviation of the
sampling distribution of that statistic.
For regression coefficients it is calculated as:
where
and
Standard Errors in Regression
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The standard error on a regression coefficients
will grow smaller both as sample size increases
and as the variance on that coefficient’s variable
increases.
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Same n, but the s.e. for the squares will be smaller
Hypothesis Test in a Regression
Example #1
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We hypothesize that IMF loans cause more political
instability. H0 for our regression slope = 0. HA 0.
We calculate a regression line. The slope coefficient on
IMF loans is 2, with a standard error of 1.
Our test statistic is a t score. It is known as a t-ratio
since it boils down to just the coefficient over the
standard error: t = (2 – 0)/ 1 = 2/1 = 2
With a level of significance = 5%, our critical values in
a t distribution with 149 d.f. are  1.98. (N = 150)
Our test statistic falls outside this range. Thus, we
reject the null hypothesis. IMF loans do seem to have
a positive effect on political instability.
Hypothesis Test in a Regression
Example #2
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We hypothesize that IMF loans cause more political
instability. H0 for our regression slope = 0. HA 0.
We calculate a regression line. The slope coefficient on
IMF loans is 2, with a standard error of 3.
Our t-ratio is 2/3 = 0.67.
With a level of significance = 5%, our critical values in
a t distribution with 149 d.f. are  1.98. (N = 150)
Our test statistic falls within this range. Thus, we fail to
reject the null hypothesis. We cannot rule out the
possibility that IMF loans have no effect on political
instability.
p values
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Many times statistical software and journal
articles will report a p value on a sample statistic.
The p value tells you the probability of
observing a sample statistic further from the null
hypothesis than the current statistic if the null
hypothesis were true.
Hypothesis testing can be done by comparing
the p value to the level of significance you want
for your test. A p value of less than 0.05 usually
means you reject the null hypothesis.
Graphical example of a p value
Example of hypothesis testing
in SPSS