Student t test vs paired t test

[#1]

I would like to compare potato yield between two varieties. These varieties planted next to each other in the field. From each variety, 12 yield samples were harvested.

I am wondering whether I should use paired t-test or 2-sample t-test for the data analysis.

Paired t-test: The degree of freedom (df) would be n-1 (12-1).

Two sample t-test: The degree of freedom (df) would be 2n-2 (24-2).

I analyzed the data using both tests:

Paired t-test showed that the two variety were statistically not different; Probability > ItI 0.1208

Two sample t-test: Showed that the varieties were statistically significant; Probability > ItI 0.0265

This shows that the 2-sample t-test is statistically more powerful.

I understand that since two sample t-test has higher degree for freedom the corresponding p value would be smaller.

Can someone provide in depth insightful about these two tests and which one I should go for?

[#2]

The choice of the test has nothing to do with the degrees of freedom or the decision from the analysis. It depends only on how the experiment was run. How were the 12 samples collected from each variety? How could you justify pairing one sample of Variety = A with one sample of Variety = B? The pair of samples would have to have existed in the same blocking structure. Each pair must exist in a different blocking structure from all other pairs.

For example, I want to test a new sole for shoes. I give a pair of shoes to 50 people. Each person gets one shoe with the old sole and one with a new sole. The pairing is by person because each subject represents a blocking structure. On the other hand, if I gave a pair of shoes with new soles to 25 people and a pair of shoes with the old sole to another 25 people, there is no inherent or justifiable pairing.

The design of the experiment determines the appropriate test to use.