Question

(23) Let X and Y represent the number of hours that a person spends cooking and doing the dishes, respectively, during a 30-day period. Assume that during a 30 -day period, the ex pected number of hours that a person spends in cooking is 20, with a variance of 40. Also assume that during a 30-day period, the expected number of hours that a person spends doing the dishes is 10 , with a variance of 25. Suppose the covariance of X and Y is 10. If 100 people are selected at random and observed for a 30-day period, what is the p that the total number of hours spent by the group doing the dishes exceeds 2900? and cooking A) 0.140 B)0.505 C)0.860 (D)).869 E) 0.875
The circled answer is wrong please show all work to arrive at a correct answer

Answer 1

here expected total hours spend in cooking and dishes per person =E(X)+E(Y)=20+10=30

and variance =Var(X)+Var(Y)+2*Cov(X,Y)=40+25+2*10=85

therefore for 100 people expected total time taken=100*30 =3000

and std deviation =sqrt(85*100)=92.195

therefore from normal approximation:

P(X>2900)=P(Z>(2900-3000)/92.195)=P(Z>-1.08)=0.860

option C is correct