(B) What is the Nonparametric Statistical Analysis? (3%)

Problem 22. (15%) An experiment was designed to study the performance of 4 different detergents for cleaning fuel injectors. The following "cleanness" readings were obtained with specially designed equipment for 12 tanks of gas distributed over 3 different models of engines:Please obtain the appropriate analysis of variance table and test at the 0.01 level of significance whether there are differences in the detergents or in the engines.

(A) Please demonstrate that the regression line will pass the centroid of all observation cases, i.e., , where α̂ and β̂ are the estimated values of α and β respectively by using least squares method., and (X̄, Ȳ) is the centroid of all observation cases. (7%)

(B) Will the situation in ( A ) happen if the constant term α does not exist any more? This is,the simple regression model is formulated as Yᵢ = βXᵢ + εᵢ  i = 1,2,....., n. (3%)

(A) What is the meaning of "Degree of Freedom"? (4%)

(C) What is the meaning of "Regression to the Mean"? (3%)

(a) f(x) =  for x = 0, 1, 2, 3, 4, 5

(b) f(x) =   for x = 0, 1, 2, 3

(c) f(x) =  for x = 3, 4, 5, 6

(d) f(x) =  for x = 1, 2, 3, 4, 5

(a) Find the value of c that makes f(x, y) a probability density function.