Question

Consider worker income from four departments: C1, C2, C3, and C4 with 3,2, 4, and 3 employees respectively: C1 C2 C3 C4 50 90 18 32 60 105 23 10 70 22 60 25 (a) Find the mean E(y) and variance V (y) where y is a single individual's income sampled at random. [10 pt] (b) Fill in the following table: C1 E(X|C) V(X|C) P(C) C2 C3 C4 where E(X|C) is the mean income in each department, V(X|C) is the variance of the income in each department, and P(C') is the probability that an individual selected at random from the population will belong to that department. [10 pt] (c) Show numerically using the above example that E(X)= E(E(X|C)) and V(X): V(E(X|C))+E(V(X|C)) [10 pt]

Answer 1

P(A random Here worker is chosen vut of 18 workers) = ElY) Hence, Mean * (50+60 +40+90+105+ (8+23+22+25+32+10+6o E Y) = 47 •083333 as : is calculatid Variance * The x (50 + Goʻ+ T090 t105* + 18*+ 23*+ 22+ 25 + 32 Vaor (4) = +10 +60 (4) = 850·74 Variance Calculation Means : of get depantment the For 3 workes of depantment C) P(A worken chosen from the is %3D E (xlcs) - f* ( *0 + 6o +1 = 60 :. E(x|) = 60 2nd Depantment , for the P(A wor Ber 2 workers of dlepantment C2) is chosen from the

E( xIC) = x (90 + 105) = 945 E(xI C) - 97.5 * For third clepartment, the 4 workers of depantment Cg) from the is chosen P(A worko + *( 18+23 + 22 + 25) E(xICs) = = 22 . E(xlcs) - 22 depantment, Fourts the For workens of dlepantment Cy) from the is chocen P(A worker E(xl C4) = 32 +10+60) - 34 E(*I Cq) :. - 34 . Calculation Of Vanian ces· : the depant ment ; For Van (x|C4) = $ *( *o + 60° + 70') -(E (x G)]* 66.66 depooitment : 2nd the For -* = (, os) - (xc]* Var (xIcs) + 105 56. 25

3rd depantment : * For the (3) x (18-2+ 25) - Var (x)G) = (xle]* = 6.5 the departrment : For Var (x[co) = fx (32*+ 10 + co") - (E(x1ca)] e 418-6666 Probabilities Calculation of .PCCI) = P C Cz) = 4. PC Cs) %3D 12 P(C4) = 12 E(E (xIc) E(x) = To show, E (E (x1c)) = E(xlC4) * PCC;) 47.08 333 E (x) E (E (XIC) = E(x). proved > Hence :.

V(x) = v[E(xIc>]+ [v(xl6] that show To Var [e (mle)] = E(} *t(c) e117. 8680556 Var (×|cr) × PCc1). E( Vao (xIc)) - (6-66 x ) + (n6-25 x*) + 56.25 x (6.5 x ) +( 416-66 x) 132. 874 : Von [e(MleJ]+ ECven (x1C)] Var N 850. 74 = 132.874 + .868 e Var (x) : Vao [E CXIO)] + E[van (xie)] Var [E («le) + E[vas (*le- Van (x) Proved Hence