Answer the Following Questions $\sum _ { i = 1 } ^ { n }$

question 77

Essay

Answer the following questions.
a. Show that $\sum _ { i = 1 } ^ { n }$ $e _ { i } = 0$
when the $e _ { i } ^ { \prime } s$
are the residuals from a simple linear regression.
b. Are the residuals from a simple linear regression independent of one another, positively correlated, or negatively correlated? Explain.
c. Show that $\sum _ { i = 1 } ^ { n } x _ { i } e _ { i } = 0$
for the residuals from a simple linear regression. [This result along with part (a) shows that there are two linear restrictions on the $e _ { i } ^ { \prime } s$
, resulting in a loss of 2 df when the squared residuals are used to estimate $\sigma ^ { 2 }$
]

---

Definitions:

Pay Differential

The variation in pay rates not explained by merit or performance, often resulting from factors like location, working hours, or job hazards.

Green-circle Rate

A situation in wage-setting where an employee's pay is below the minimum of the range set for their job class, often due to an anomaly or an exception in the salary structure.

Rank-and-file Adjustment

Changes or modifications made to the conditions or ranks of the general workforce, not the leadership or management.

Uniformed Services Employment and Reemployment Rights Act

A federal law that provides job protection and rights for military service members upon their return to civilian employment.