I'm running the example of the R-intro manual:
A = c(79.98, 80.04, 80.02, 80.04, 80.03, 80.03, 80.04, 79.97, 80.05, 80.03, 80.02, 80.00, 80.02)
B = c(80.02, 79.94, 79.98, 79.97, 79.97, 80.03, 79.95, 79.97)
t.test(A, B)
Which produces the following result:
Welch Two Sample t-test
data: A and B
t = 3.2499, df = 12.027, p-value = 0.006939
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
0.01385526 0.07018320
sample estimates:
mean of x mean of y
80.02077 79.97875
The question is: if the difference of means is contained within the confidence interval (80.02077-79.97875=0.04202 and 0.01385526<0.04202<0.07018320) why does it conclude that the alternative hypothesis is true and not that the null hypothesis is true?