Consider the shoes data from the MASS package:
data(shoes, package="MASS")
shoes
## $A
## [1] 13.2 8.2 10.9 14.3 10.7 6.6 9.5 10.8 8.8 13.3
##
## $B
## [1] 14.0 8.8 11.2 14.2 11.8 6.4 9.8 11.3 9.3 13.6
We shall do
Compare two shoe types with a \(t\)-test:
with(shoes, t.test(A, B))
##
## Welch Two Sample t-test
##
## data: A and B
## t = -0.36891, df = 17.987, p-value = 0.7165
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.745046 1.925046
## sample estimates:
## mean of x mean of y
## 10.63 11.04
The test is misleading because observations are paired. A better alternative is to make a paired \(t\)-test:
with(shoes, t.test(A, B, paired=T))
##
## Paired t-test
##
## data: A and B
## t = -3.3489, df = 9, p-value = 0.008539
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.6869539 -0.1330461
## sample estimates:
## mean of the differences
## -0.41