Agricultural Statistics

Derivation of regression line of $Y$ on $X$:

• $Y = X + 7.3$ has a slope of $1$, which does not match $b_{yx}$.
• The regression coefficient of $Y$ on $X$ is $b_{yx} = \rho \frac{\sigma_y}{\sigma_x} = 0.84 \times \frac{6.4}{8.4} = 0.84 \times \frac{16}{21} = 0.64$. The regression equation is $(Y - \mu_y) = b_{yx}(X - \mu_x) \implies Y - 33 = 0.64(X - 18.4) \implies Y - 33 = 0.64X - 11.776 \implies Y = 0.64X + 21.224 \approx 0.64X + 21.22$.
• $Y = 0.64X + 10.61$ results from subtracting rather than adding the constant.
• $Y = 0.4X + 12.82$ uses an incorrect slope.

Calculation of quartiles for $n = 20$ sorted observations:

• The sorted values are: $1, 3, 5, 5, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10$.
• The first quartile $Q_1$ lies between the $5^{\text{th}}$ observation ($7$) and $6^{\text{th}}$ observation ($8$), yielding $Q_1 = 7.5$.
• The median $Q_2$ is the average of the $10^{\text{th}}$ ($8$) and $11^{\text{th}}$ ($8$) observations, giving $Q_2 = 8$.
• The third quartile $Q_3$ lies between the $15^{\text{th}}$ observation ($9$) and $16^{\text{th}}$ observation ($9$), giving $Q_3 = 9$. Hence, $Q_1 = 7.5, Q_2 = 8, Q_3 = 9$.

Minimal property of standard deviation:

• Standard deviation is the minimum, never the maximum.
• The root mean square deviation about any arbitrary point $A$ is $S = \sqrt{\frac{1}{n} \sum (X_i - A)^2}$. By algebraic properties, $S^2 = \sigma^2 + (\bar{X} - A)^2$. Since $(\bar{X} - A)^2 \ge 0$, $S^2$ attains its unique absolute minimum when $A = \bar{X}$, meaning standard deviation is the least possible value of root mean square deviation.
• Standard deviation equals r.m.s.d. only when deviations are taken from the arithmetic mean, not universally.
• False proportional relationship.