Statistical Inference and Hypothesis Testing

1
Statistical Inference and Hypothesis Testing

• Political Science 102
• Introduction to Political Inquiry
• Lecture 18

2
Hypothesis Testing Procedures
3
Factors Influencing Statistical Significance
4
Errors in Hypothesis Testing

• In selecting threshold for judging statistical
  significance, we should balance the costs of type
  I and type II errors
• Standard scientific practice is to set threshold
  for rejecting H0 at 95 (i.e. .05 level)
• Assumes type II errors are preferable (a
  conservative standard)
• Other thresholds may be more appropriate for
  different problems or applications

5
Is it a Toss Up?
6
Hypothesis Tests and Sampling Distributions

• Hypothesis tests depend on estimated confidence
  intervals for the parameter
• Confidence interval depends on sampling
  distribution of the statistic
• Distributions of variables may differ
  substantially and idiosyncratically
• Seems to make problem intractable
• But we want the sampling distribution of the mean
  (or some other estimator statistic)
• NOT sampling distribution of parent variable

7
Calculating Confidence Intervals

• Central Limit Theorem tells us
• Sum of a set of random variables approaches a
  normal distribution as n approaches 8
• Normal distribution is the bell curve
• The mean (and other sample statistics) are sums
  of random variables
• We can rely on the normal distribution to test
  hypotheses about the value of population
  parameters based on information from sample
  statistics

8
An Example of the Central Limit Theorem

• Expected value of a fair coin toss0.5
• Heads1, Tails0
• Distribution of coin toss result is NOT normal
• Result is 0 or 1
• Bernouli distribution
• But mean value of coin toss is normally
  distributed!

9
Statistical Inference
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Hypothesis Tests

• Our theories give us hypotheses about population
  parameters
• µX gt µY
• The value of X has a positive impact on the value
  of Y
• We can estimate sample statistics
• The mean of X and Y in our sample
• We need a way to assess the validity of
  statements about population parameters
• We can rely on our sample estimators and their
  variance to use probability theory to test such
  statements.

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Z Scores Hypothesis Tests

• We know that N( µX , sX )
• Subtracting µX from both sides, we can see that
• - µX N( 0 , sx )
• The if we divide by the standard deviation we can
  see that
• - µX / sX N( 0 , 1 )
• This variable is a z-score based on the
  standard normal distribution.
• 95 of cases are within 1.96 standard deviations
  of the mean.

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Z-Scores Hypothesis Tests

• We can use this to test the hypothesis that µX is
  equal to a particular value given our sample mean
• Same logic will be used later to test for a
  relationship between X and Y
• If - hypothesized value / sX gt 1.96 then
• there is a 95 chance that µX ? hypothesized
  value

13
Hypothesis Testing

• How do you test hypotheses with statistics?
• Comparing the means of two groups
• Consider an experiment
• Research hypothesis
• Null hypothesis

-
-
X1 ? X2
-
-
X1 X2
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Hypothesis Testing

• Hypothesis College students read fewer political
  news stories per week than other voting-age
  citizens.
• (college mean) 5
• ? (population mean) 10
• ? (college std. dev.) 2
• n 25

_
(X ?)
z
__________
(? / vn)
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Hypothesis Testing

• Hypothesis College students read fewer political
  news stories per week than other voting-age
  citizens.
• (college mean) 5
• ? (population mean) 10
• ? (college std. dev.) 2
• n 25

-12.5
z
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Hypothesis Testing

• Hypothesis College students read fewer political
  news stories per week than other voting-age
  citizens.
• 95 confidence
• z critical 1.96

-12.5
z
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Z-Scores and t-scores

• Problem with z-scores for testing hypothesis is
  that we generally do not know the true variance
  of X
• Obvious solution is to substitute the SAMPLE
  variance of x
• Problem The sample mean of X divided by the
  sample variance is the ratio of two random
  variables, and this will not be normally
  distributed
• Fortunately, an employee of Guiness Brewery
  figured out this distribution in 1919
• The statistic is called Students t, and the
  t-distribution looks similar to a normal
  distribution

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The t-statistic

• More generally X-bar / sX-hat t(n-k)
• k is the of parameters estimated
• Note the addition of a degrees of freedom
  constraint
• Relates to how much independent information we
  have
• Lose one piece of independent information each
  time we estimate a parameter (like the variance
  of X)
• Thus the more data points we have relative to the
  number of parameters we are trying to estimate,
  the more the t distribution looks like the z
  distribution.
• When Ngt100 the difference is negligible

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A Real World ExampleFeminists vs.
Environmentalists
. sum v5072 v5059 Variable Obs
Mean Std. Dev. Min
Max ---------------------------------------------
------------------------ v5072 1043
66.0326 20.19167 0 100
v5059 1028 56.33171 21.68065
0 100
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A Real World ExampleFeminists vs.
Environmentalists
One-sample t test --------------------------------
---------------------------------------------- Var
iable Obs Mean Std. Err. Std.
Dev. 95 Conf. Interval ---------------------
--------------------------------------------------
------ v5059 1028 56.33171 .6762009
21.68065 55.00482 57.65861 --------------
--------------------------------------------------
-------------- mean mean(v5059)
t 9.3637 Ho
mean 50
degrees of freedom 1027 Ha mean lt 50
Ha mean ! 50 Ha
mean gt 50 Pr(T lt t) 1.0000 Pr(T gt
t) 0.0000 Pr(T gt t) 0.0000
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A Real World ExampleFeminists vs.
Environmentalists
Paired t test ------------------------------------
------------------------------------------ Variabl
e Obs Mean Std. Err. Std. Dev.
95 Conf. Interval ---------------------------
--------------------------------------------------
v5059 1021 56.34574 .6790026
21.69623 55.01334 57.67814 v5072
1021 66.0431 .6326196 20.21415
64.80171 67.28448 ----------------------------
-------------------------------------------------
diff 1021 -9.697356 .663641
21.20538 -10.99961 -8.395098 -----------------
--------------------------------------------------
----------- mean(diff) mean(v5059 - v5072)
t -14.6124 Ho
mean(diff) 0
degrees of freedom 1020 Ha mean(diff) lt
0 Ha mean(diff) ! 0 Ha
mean(diff) gt 0 Pr(T lt t) 0.0000 Pr(T
gt t) 0.0000 Pr(T gt t) 1.0000