(Requires probability.) In an SRSWR, a population unit can appear in the sample anywhere between 0 and n times. Let Qi = number of times unit i appears in the sample, and ˆt = N n N i=1 Qiyi. a Argue that the joint distribution of Q1,Q2, . . . ,QN is multinomial with n trials and p1 =p2= · · · =pN =1/N. b Using (a) and properties of the multinomial distribution, show that E[ˆt ]=t. c Using (a) and properties of the multinomial distribution, find V[ˆt ].

Martha has a deck of cards. she has lost some of the cards, and now the deck only contains nine spades, eleven diamonds, eight clubs, and twelve hearts. martha predicts that whenever she draws a card from the deck without looking, she will draw a club one-fifth of the time. which activity would best allow martha to test her prediction? a. randomly draw a card from the box and see if it is a club. b. randomly draw a card. then, continue to draw another card until all eight clubs are drawn. c. randomly draw and replace a card 120 times. then, observe how close to 30 times a club is drawn. d. randomly draw and replace a card 100 times. then, observe how close to 20 times a club is drawn.