Question

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one of the top smartphones to get this holiday season, and to check 5.) The Apple iPhone 5S is clearly random sample of 41 technology bloggers. Of the 41 sampled, 27 believe the Apple iPhone 5S to be the top smartphone on the market. We are interested in constructing an 88% confidence interval for the true population proportion of technology bloggers that prefer the Apple iPhone 5S we asked a a) Describe p' in words, in the context of the problem. Then, describe its distribution & explain why b) Calculate, by hand, the error bound E & an 88% confidence interval for the true population proportion. Also, interpret the meaning of the confidence interval in the context of the problem. c) How many total bloggers must be sampled in order for the error bound to decrease by 0.05?

Answer 1

(A) Single Powpontion: Let be the Proportion of membeas with a chanaet A in a populahion. Liet chonaterishic A in a Yandom kamples of Size n drawn WR be he number of mnembens witF

population Bin(n, P)and p= Then - ample puopostion of characteishe A . me stoktic p= mean pand aniance Pl-p)- has -tenialTheogp-E()_. NCo), n lane PC-P) BycT VanF Under the null hypothesis H: pzp, , the dest zlakistic NCo,4) Z Pli- Po) xejested against H,: p+p. atlevdl of Triencfire Ho:p=p. is SnifiCanee Value of z, he p-vaue is 2 P[ z> 1z.1]. Zo is the dbgerv." Confidence Internvel: F6 lange n, the eshimate of PCE)md we.aso have Jñ(p-P) (TN An approximate doo(i-K)'. C.I ísgven by - - X - g-i)d Hence fon he given Sample , the obLenved Valueg of CB-TE + ]wil be onfidence inteantak 0with onfidence Koeficient apporaeimatek equal

* 4 enstaad of p' dhaoushut m gonnn usC The ans wer, * a) 7/41 Bample pncpontion of blogger Who brebers fhe atple Phone 5S N(, n=4.1 X-no of llog gen nelen atple i Phone 5S X~T(41, b) Bin popopulation popuition Using eLT X-E(X) N(0,1) X/n-P (0,4) X-np mP(4p) E = TapPC1) A=0.12 To o6= 1. 55 O.65% X0.342 1.55 41 - 0.414 0.444 ,0658+0.114) Hen ee 88 e S. given by (0658-0 (o.544, 0772) Tnlen Pnetchun Protontion cb lloggery Who Pne beas iPhune 5S An fenval will contoin rue Pobulationlluith 887 Conbidag Thi c) Given O.444-0.05 E= Ta n 1.55 X0.6 59 XO 32 11.48 0.064 n 132hacens need to le samhled