The Scientific Study of Politics (POL 51)

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The Scientific Study of Politics (POL 51)

• Professor B. Jones
• University of California, Davis

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Fun With Numbers

• Z-scores
• R code (wrt hw. 4)

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Normal Distribution

• Normal Distribution and areas under it.
• 68-95-99.7 Percent Rule
• In a normal distribution, about 68 percent of the
  observations will fall within about /- 1
  standard deviation...
• A Picture

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Area (with some added stuff)
http//members.aol.com/svennord/ed/normal.htm
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What do we know?

• Area is useful to determine probabilities.
• Fun with Numbers
• Gas Prices (Lets take a sidetrip)
• What are some research issues when looking at
  financial data over time?
• Inflation!
• 2007 dollars vs. 1990 dollars
• CPI 2007 Price1990 Price(2007 Price/1990 Price)

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Visualizing Data is FUNdamental
Unadjusted CPI-Adjusted
CPI-Adjusted w/o 05/06)
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Histograms
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Using z-scores

• Taking advantage of the normal distribution
• Area under the normal is probability area.
• Probabilities must sum to 1.
• Full density under normal is 1.
• Since its symmetric, we know the probability of
  being above the mean is .50 (ditto on below)

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Standard Normal Distribution

• N(0,1)
• Easy to compute
• When Xmean, z0.
• Metric of z-score standard deviations from the
  mean.
• Thus, if z1, X is 1 s.d. above the mean.
• NOW since we know the 68-95-99.7 Rule, we can
  identify probs.

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Getting Gas

• Lets look at the adjusted gas prices.
• Means
• 2006 2.57 (.30) 1999 1.37 (.15)
• 2005 2.34 (.32) 1998 1.27 (.04)
• 2004 1.98 (.15) 1997 1.51 (.04)
• 2003 1.71 (..09) 1996 1.54 (.08)
• 2002 1.51 (.13) 1995 1.47 (.06)
• 2001 1.62 (.20) 1994 1.46 (.07)
• 2000 1.74 (.11) 1993 1.49 (.03)
• 1992 1.56
  (.07)
• 1991 1.62
  (.05)
• 1990 2.00
  (.07) small n
• (Anything interesting here?)
  

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Compute a z-score

• Mean adjusted price 1.68 (.37)
• To derive z-score for any year, substitute a
  value X into ?
• Suppose X1.68?
• Z(1.68-1.68)/.370
• The mean is normalized to 0.
• 1 s.d. above mean? 1.68.372.05
• Z(2.05-1.68)/.371
• The metric of z is in standard deviations.

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Standardizing X allows us to use z
distribution.
The Most Average Price
z Week Year ---------------------
----------------- 1.680374 -.009361 Feb
12 2001 1.681257 -.0069663 Nov 03
2003 1.681329 -.0067707 Apr 24 2000
1.682352 -.0039966 Aug 04 2003
1.683292 -.001449 Jun 03 1991
1.684771
.0025612 Feb 04 1991 1.68625
.0065716 May 27 1991 1.688924
.0138213 Oct 27 2003 1.689519
.0154355 Apr 17 2000 1.69062
.0184197 Sep 24 2001 --------------------
------------------
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The 10 Most Above Average
Price Z Week Year
-------------------------------------
2.947 3.424879 May 15 2006 2.973
3.495373 Jul 10 2006 2.989
3.538755 Jul 17 2006 3
3.56858 Aug 14 2006 ---------------------
---------------- 3.003 3.576713 Jul
24 2006 3.004 3.579425 Jul 31
2006 3.021628 3.62722 Oct 03 2005
3.038 3.67161 Aug 07 2006
3.049491 3.702766 Sep 12 2005
3.167136 4.021741 Sep 05 2005
-------------------------------------
The 10 Most Below Average Price
Z Week Year -------------------------
------------- 1.096723 -1.59183 Feb 22
1999 1.103978 -1.572159 Mar 01 1999
1.111233 -1.552488 Feb 15 1999
1.113652 -1.545931 Mar 08 1999
1.120907 -1.52626 Feb 08 1999
--------------------------------------
1.123325 -1.519703 Feb 01 1999
1.13058 -1.500032 Jan 04 1999
1.131789 -1.496754 Jan 25 1999
1.137835 -1.480361 Jan 11 1999
1.141463 -1.470526 Jan 18 1999
--------------------------------------
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Finding Probabilities

• What is the probability of a z gas price of 2.50
  or higher?
• The z-score is 2.22.
• In the z-distribution, if gas prices were truly
  normally distributed, a score this high or higher
  has a probability of occurring of .013, or about
  1.3. Its an unlikely event.
• How computed? 1-.9868 gives area above (consult
  standard normal)

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Finding Probabilities

• What is the probability of a z gas price being
  between 1.75 and -1.75
• P(above).04 P(below).04
• Therefore, P(in between)1-.08 .92
• The upper tail is .04 the lower tail is .04
• Any probability calculation is this
  straightforward.

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For Baseball Fans

• http//alexreisner.com/baseball/stats/

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Issues

• The gas price example is pedagogical.
• Serious analysis of gas-pricing effects would
  require much more sophisticated statistical
  techniques.
• z is useful to compare observations from
  historical eras or across disparate cases.