Paired t-test

Two data samples are said to be paired (or matched) if they come from repeated observations of the same subject. Here, we assume that the data populations follow the normal distribution.

Using the paired t-test, we can obtain an interval estimate of the difference of the population means. There must be equal numbers of elements in both sets.

The t.test() function can be used to perform paired t-tests by specifying the argument paired = TRUE.

Barley Yield Example

In the built-in data set named immer, the barley yield in years 1931 and 1932 for the same fields is recorded. Fertilizer treatments were applied in the interim. The study aimed to determine whether the treatment was effective.

The yield data are presented in the data frame columns Y1 and Y2.

library(MASS) # load the MASS package
head(immer)
  Loc Var    Y1    Y2
1  UF   M   81.0  80.7
2  UF   S  105.4  82.3
...

Assuming the data in immer follows the normal distribution, we want to find the 95% confidence interval estimate of the difference between the mean barley yields.

We apply the t.test() function to compute the difference in means of the matched samples. As it is a paired test, we set the paired argument as TRUE.

attach(immer)
t.test(Y1, Y2, paired = TRUE)
    Paired t-test

data:  Y1 and Y2
t = 3.324, df = 29, p-value = 0.002413
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
  6.121954 25.704713
sample estimates:
mean of the differences 
             15.91333

Between years 1931 and 1932 in the data set immer, the 95% confidence interval of the difference in means of the barley yields is the interval between 6.122 and 25.705.

We conclude that the fertilizer treatments were successful in improving barley yields.