In a layout of a randomized block design with 7 treatments, each... | Agrigyan

Derivation of Standard Error of Difference:
Number of replications $r = 4$.

• Correct: The standard error of the difference between two treatment means $\bar{y}_i$ and $\bar{y}_j$ is given by:
  $$\text{SE}_d = \sqrt{\text{Var}(\bar{y}_i - \bar{y}_j)} = \sqrt{\frac{\text{MSE}}{r} + \frac{\text{MSE}}{r}} = \sqrt{\frac{2\text{MSE}}{r}}$$
  Substituting $r = 4$:
  $$\text{SE}_d = \sqrt{\frac{2\text{MSE}}{4}} = \sqrt{\frac{\text{MSE}}{2}}$$
• Incorrect: $\sqrt{\text{MSE}/4}$ is the standard error of a single treatment mean $\text{SE}(\bar{y}_i) = \sqrt{\text{MSE}/r}$.
• Incorrect: 5 is the number of treatments, which does not appear in the numerator of $\text{SE}_d$.
• Incorrect: Invalid expression.

Analysis of regression properties under transformation:

• Incorrect: Adding a constant does not increase the slope.
• Incorrect: Division by 35 occurs only if variables are scaled by multiplying by 35.
• Incorrect: Subtraction does not affect the slope.
• Correct: Regression coefficients ($b_{yx} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$) are strictly independent of the change of origin. Adding any constant (here 35) to all values of $X$ and $Y$ shifts the mean but leaves deviations $(X - \bar{X})$ and $(Y - \bar{Y})$ completely unchanged. Hence, the regression coefficient remains exactly the same as the original.

Mathematical solution and official examination key alignment:
Given $P(X \lt 35) = 0.1003$ and $P(X \lt 80) = 0.8997$.
From the standard normal table, the cumulative probability $0.8997$ corresponds to $Z = 1.28$, and $0.1003$ corresponds to $Z = -1.28$.
Theoretical symmetry gives $\sigma = \frac{80 - 35}{2 \times 1.28} = 17.58$ and $\mu = 57.5$.
Under the official NTA answer key formulation for this test paper:

• Incorrect: Does not match official key.
• Incorrect: Does not match official key.
• Incorrect: Does not match official key.
• Correct: Option 4 (mean = 52.5 and sd = 21.48) is the declared official answer key by the exam authority.