For any two positive quantities, the fundamental inequality of averages is:

For the two regression coefficients $b_{yx}$ and $b_{xy}$:

Substitute $X = 20$ into the estimated regression equation $\hat{Y} = 100 + 7X$:

Analysis of degrees of freedom in RBD:
Total number of experimental units is $N = n \times r$.

• Incorrect: $r - 1$ is the degrees of freedom for Replications (Blocks).
• Incorrect: $n - 1$ is the degrees of freedom for Treatments.
• Incorrect: $n \times r - 1$ is the Total degrees of freedom.
• Correct: In a Randomized Block Design (RBD) with $n$ treatments and $r$ replications:
  $$\text{Total df} = nr - 1$$
  $$\text{Block df} = r - 1$$
  $$\text{Treatment df} = n - 1$$
  $$\text{Error df} = (nr - 1) - (r - 1) - (n - 1) = (n - 1)(r - 1)$$

Analysis of experimental designs:

• Correct: A Completely Randomized Design (CRD) utilizes only two basic principles of experimental design (randomization and replication, without local control). It is valid only when experimental units are homogeneous (e.g., in laboratory, pot culture, or greenhouse experiments).
• Incorrect: CRD handles missing units flexibly, but the fundamental suitability criterion is unit homogeneity.
• Incorrect: When units are heterogeneous, blocking designs (RBD or LSD) are required.
• Incorrect: Different slopes introduce a directional fertility gradient, demanding RBD.