Student's t test 3

The paired t test is used when two sets of values are related, for example because each of a pair of measurements was made on the same subject.

In this case, it is the mean of the difference between the two values that is distributed according to the t distribution.

This example is from Dalgard.

pre = c(5260,5470,5640, 6180,6390,6515,6805, 7515,7515,8230,8770) post = c(3910,4220,3885, 5160,5645,4680,5265, 5975,6790,6900,7335) plot(pre,post,pch=16, col='blue',cex=2) diff = post-pre

Not only are the values correlated, but the difference is always negative:

> diff [1] -1350 -1250 -1755 -1020 [5] -745 -1835 -1540 -1540 [9] -725 -1330 -1435

t.test(pre,post,paired=T)

> t.test(pre,post,paired=T) Paired t-test data: pre and post t = 11.9414, df = 10, p-value = 3.059e-07 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 1074.072 1566.838 sample estimates: mean of the differences 1320.455

We can do the test by hand, as follows:

> mean(diff) [1] -1320.455 > sd(diff) [1] 366.7455 > x = sd(diff)/sqrt(10) > x [1] 115.9751 > abs(mean(diff))/x [1] 11.38567

The question now is, what fraction of the values from the t-distribution with df = 10 are greater than 11.39?

S = seq(0,1,by=0.001) w = rt(1000000,df=10) y = quantile(w,S) round(tail(y))

> round(tail(y)) 99.5% 99.6% 99.7% 3 3 3 99.8% 99.9% 100.0% 4 4 11

The short answer: not very many!