I do a lot of statistical work in my day job, but I don’t use basic probability all that much. So hopefully I’m not screwing this up. But I’d like to do something unusual for a blog, and apply some formal probability theory to the question of whether growth rates of real GDP per capita for Democrats and Republicans are different. Apologies to those that find this boring. You may want to simply skip down to the bottom.
The two-sample t test allows you to test whether the sample means of two different samples with two different standard deviations are different. It does not require a large number of observations, but does assume that the two samples from which the populations are drawn are normal.
Say we want to test a null hypothesis,
H0: Mean of Democratic Growth rate = Mean of Republican Growth rate, versus
H1, Mean of Democratic Growth rate Mean of Republican Growth rate
If we look at the growth rate in quarterly real GDP per capita for Reps and Dems separately, we find that:
Republican count = 136
Republican mean = 0.38%
Republican standard deviation = 0.98%
Democratic count = 80
Democratic mean = 0.72%
Democratic standard deviation = 0.80%
(All of this derived from data in NIPA Table 7.1.)
The degrees of freedom in this case = 136 + 80 – 2 = 214
The t distribution, at 214 degrees of freedom at the alpha = 0.01 level of significance is equal to 2.58. This means that the null hypothesis should be rejected if the t statistic is exceeds 2.58, or falls below –2.58.
But the t statistics is equal to 2.71. Therefore, the null should be rejected. It would appear that Dems and Reps do not have the same growth rate. At the 1% level of significance.
One cool thing about this test… you don’t actually need many observations. I’m checking my old college Probability book right now, Larsen and Marx, “An Introduction to Mathematical Statistics and its Applications”, second ed., pp 364 – 367, and they run a test with 8 observations in one sample and 10 in another.
So I’m going to try this again, just looking at Presidents… I’m looking now at real GDP per capita by administration.
Republican count = 5
Republican mean = 1.514%
Republican standard deviation = 0.599%
Democratic count = 3
Democratic mean = 2.705%
Democratic standard deviation = 0.693%
If I’m doing things right, the t statistics is 2.47.
In this case, the degrees of freedom = 5 + 3 – 2 = 6.
Now, this isn’t a lot of observations, and we can’t reject the null hypothesis at alpha =0.01. But… we can reject the null hypothesis at alpha = 0.05, since the t with six degrees of freedom and an alpha of 5% is rejected if 2.45.
So unless I’m making a serious mistake, the growth rates produced by the two parties do not appear to be the same, whether we assume quarterly (the probability they’re the same is less than 1%) or for entire administrations (the probability they’re the same is less than 5%).