## Tuesday, March 19, 2013

### Statistics Question

I think I know enough about statistics to be trusted with this, but I suspect some of you who do this day in and day out can check my method.

A particular state's murder rate per 100,000 for 1999 through 2010:

 1999 6.6 2000 6.2 2001 6.6 2002 5.8 2003 5.1 2004 6.1 2005 6.9 2006 6.3 2007 6.2 2008 7.7 2009 6.5 2010 7

There was a change in state law for which the first full year was 2008.  The average for 1999-2007 was 6.2; for 2008-2010: 7.1 (rounded).  Std. dev. is 0.494 and 0.492, respectively.  Using Student's T (appropriate because n is so small for both year ranges), for a 95% confidence interval, .380 and 1.222 respectively.  This indicates that the 13.9% increase in murder rates for the years 2008-2010 is not statistically significant; the uncertainty range for 1999-2007 is 5.82 to 6.58; for 2008-2010, 5.84 to 12.91.  In short, it is impossible to determine at the 95% confidence level if the change in law caused the changed the increase in murder rate.  You have to get down to the 80% confidence level before you get statistical significance -- which isn't terribly impressive.

Interestingly: enough the firearms homicide rate, which may or may not include justifiable homicides, rose 25% -- far more than the murder rate.

Do I have this right?

UPDATE: I ran it by a nationally known economist who confirms that the methodology is right.

#### 1 comment:

Basil said...

Clayton,

I haven't checked your means and std deviations, but taking them as you calculated them, I plugged them into a stat calculator and got this:
====================
Null hypothesis: Difference of means = 0

Sample 1:
n = 9, mean = 6.2, s.d. = 0.494
standard error of mean = 0.164667
95% confidence interval for mean: 5.82028 to 6.57972

Sample 2:
n = 3, mean = 7.1, s.d. = 0.493
standard error of mean = 0.284634
95% confidence interval for mean: 5.87532 to 8.32468

Test statistic: t(10) = (6.2 - 7.1)/0.3292 = -2.7339
Two-tailed p-value = 0.02105
(one-tailed = 0.01053)
============================

This is showing a statistically significant difference at 95 percent. Doing a little sensitivity analysis, the difference becomes statistically significant when the mean of sample 2 gets to 6.933. The critical t here is "2.228" and according to the above, we're in the tail at -2.733.

Basil