STT 200 LECTURE 5 SECTION 23 24 RECITATION

STT 200 – LECTURE 5, SECTION 23, 24 RECITATION 11 (3/26/2013) TA: Zhen Zhang zhangz 19@stt. msu. edu Office hour: (C 500 WH) 3 -4 PM Tuesday (office tel. : 432 -3342) Help-room: (A 102 WH) 9: 00 AM-1: 00 PM, Monday 1 Class meet on Tuesday: 12: 40 – 1: 30 PM A 224 WH, Section 23 1: 50 – 2: 40 PM A 234 WH, Section 24

MAIN GOALS 2

DATA q Here are data from a population of 400 people, indicating whether they do ("Yes") or don't ("No") have wireless internet service at home. Please copy the following chunk and paste in R. haswi <- c("Yes", "No", "Yes", "Yes", "Yes ", "No", "Yes", "No", "No", "Yes", "No", "Yes", "Yes ", "No", "Yes", "Yes", "No", "No", "Ye s", "Yes", "No", "No", "Yes", "Yes", "No", "No", "Yes", "No", "No", "Yes", "Yes", "Yes", "No", "Yes", "No", "Yes", "No", "Yes", "Yes", "Yes", "No", "Yes", "No", "Yes", "Yes", "No", "Yes", "No", "Yes", "Yes", "No", "Yes", "No", " Yes", "No", "Yes", "Yes", "Yes", "No", "Yes", "No", "Yes", "Yes", " No", "Yes", "Yes", "Yes", "No", "Yes", "No", "Yes", "No", "Yes", "Y es", "No", "Yes", "No", "Yes", "No", "Yes", "Ye s", "No", "Yes", "Yes", "Yes", "No ", "Yes", "No", "Yes", "No", "Yes", "No", "Yes", "No", "Yes", "No", "Ye s", "Yes", "No", "Yes", "No", "Yes", "No", "Yes ", "No", "Yes", "No", "Yes", "Yes", "No", "No", "Yes", "Yes", "No", "Yes", "No", "No", "Yes", "No ", "No", "Yes", "No", "Yes", "Yes", "No", "Y es", "No", "Yes", "No", "Yes", "No", 3 "No", "Yes", "Yes", "No", "Yes", "No", " Yes", "No", "Yes", "No")

DATA q Here is a table of integers between 1 and 400 chosen at random. R chuck: rd <- c(92, 149, 41, 310, 307, 130, 296, 130, 77, 399, 212, 301, 25, 177, 313, 147, 298, 160, 354, 20, 199 , 191, 104, 164, 216, 399, 25, 99, 28, 91, 211, 357, 350, 301, 39, 372, 61, 67, 304, 333, 174, 321, 191, 157, 31 6, 172, 5, 277, 78, 396, 208, 126, 162, 311, 17, 287, 138, 160, 124, 266, 177, 209, 361, 41, 398, 9, 79, 299, 2 57, 315, 40, 278, 2, 225, 206, 383, 254, 74, 335, 159, 37, 360, 9, 393, 143, 246, 305, 152, 90, 312, 208, 172, 117, 277, 93, 399, 226, 8, 231, 386, 136, 75, 38, 56, 37, 267, 381, 63, 52, 231, 287, 94, 50, 77, 179, 337, 387, 318, 112, 219, 17, 356, 77, 183, 259, 258, 141, 198, 30, 36, 61, 306, 65, 330, 161, 348, 19, 20, 61, 275, 365, 241, 115, 4, 338, 205, 108, 241, 190, 374, 323, 243, 146, 318, 217, 375, 267, 44, 373, 185, 341, 283, 200, 17 8, 266, 390, 232, 263, 386, 36, 270, 50, 315, 83, 90, 281, 260, 41, 305, 136, 116, 185, 25, 338, 4, 367, 296, 1 83, 103, 290, 208, 170, 143, 158, 198, 132, 155, 144, 26, 104, 281, 150, 240, 68, 67, 339, 389, 345, 141, 268, 349, 99, 147, 65, 170, 375, 317, 251, 185, 278, 80, 250, 4, 378, 175, 130, 359, 319, 400, 59, 166, 147, 130, 1 07, 123, 304, 234, 41, 20, 165, 96, 115, 272, 149, 142, 75, 262, 235, 106, 107, 354, 362, 2, 81, 89, 309, 371, 10, 282, 203, 156, 386, 130, 252, 26, 387, 143, 237, 183, 328, 306, 27, 187, 310, 321, 183, 109, 198, 20 0, 281, 70, 394, 378, 203, 42, 34, 318, 156, 255, 354, 53, 196, 20, 382, 97, 292, 188, 179, 69, 151, 14, 348, 3 11, 389, 298, 399, 104, 300, 243, 163, 316, 328, 65, 167, 200, 301, 305, 27, 176, 69, 301, 188, 192, 242, 350, 92, 86, 42, 373, 195, 118, 64, 289, 329, 131, 156, 252, 169, 299, 191, 302, 19, 83, 220, 326, 229, 285, 267, 3 51, 333, 101, 128, 146, 307, 304, 245, 264, 149, 163, 353, 276, 296, 243, 8, 127, 31, 210, 263, 384, 176, 125, 275, 76, 45, 60, 59, 143, 324, 281, 376, 298, 54, 62, 170, 295, 293, 27, 183, 126, 375, 21, 294, 242, 364, 145, 138, 52, 267, 26, 308, 391, 352, 78, 98, 211, 174, 277, 176, 74, 295, 64, 315, 171, 135, 159, 111, 79, 34 8, 88, 23, 348, 111, 188, 16, 152, 212, 104, 349, 14, 272, 209, 73, 238, 146, 50, 113, 103, 204, 389, 158, 260, 344, 207, 329, 184, 250, 38, 231, 292, 300, 34, 170, 343, 233, 275, 14, 15, 244, 104, 96, 234, 297, 113, 270, 369, 202, 37, 310, 294, 64, 183, 253, 299, 287, 225, 166, 260, 125, 198, 2, 180, 219, 117, 358, 191, 301, 310, 254, 230, 296, 2, 134, 67, 186, 265, 161, 130, 257, 166, 339, 332, 137, 61, 340, 16, 212, 209, 42, 315, 8, 269, 68, 389, 316, 355, 62, 51, 64, 388, 260, 319, 244, 116, 265, 169, 153, 147, 170, 59, 329, 261, 384, 272, 367, 177, 217, 278, 266, 307, 182, 225, 80, 264, 342, 280, 350, 366, 280, 156, 323, 208, 110, 37, 266, 260, 5 9, 33, 314, 80, 185, 87, 228, 246, 61, 369, 60, 119, 179, 326, 223, 128, 62, 98, 130, 283, 328, 225, 398, 3, 138, 140, 84, 381, 234, 131, 364, 294, 59, 343, 126, 93, 14, 204, 50, 35, 161, 15, 142, 275, 72, 254, 194, 3 09, 115, 344, 378, 267, 23, 111, 168, 334, 92, 213, 1, 181, 246, 336, 52, 82, 4, 115, 286, 3, 87, 121, 84, 281, 4 181, 58, 372, 232, 30, 279, 258, 154, 37, 6, 113, 125, 317, 123, 198, 25, 388, 268, 106 )

PROBLEMS 5

SIMULATION 6

PROBLEMS 7

PROBLEMS 8

PROBLEMS 9

APPENDIX q R codes for the problems. # prob 4: n <- 25; p <- 0. 5575 ( sdphat <- sqrt(p*(1 -p)/n) ) # prob 5: ( pnorm(p+0. 1, p, sdphat) - pnorm(p-0. 1, p, sdphat) ) # prob 6: n 2 <- 100 ( sdphat 2 <- sqrt(p*(1 -p)/n 2) ) ( pnorm(p+0. 1, p, sdphat 2) - pnorm(p-0. 1, p, sdphat 2) ) # comparison of n=25 and n=100 vec <- seq(0. 01, 0. 99, length=1000) par(yaxt='n', mar=c(4, . 3, . 3 )) plot(dnorm(vec, p, sdphat 2)~vec, type='n', ylab=' ', xlab=expression(hat(p))) grid(col='gray 80') lines(dnorm(vec, p, sdphat)~vec, lty=1, lwd=2) lines(dnorm(vec, p, sdphat 2)~vec, lty=2, lwd=2) abline(v=p, col='red', lty=2 ) text(x=p, y=0, labels=paste ("p =", round(p, 4)), col='red') legend('topleft', legend=c(paste('N(', round(p, 4), ', ', round(sdphat, 4), '), n=25', sep=''), paste('N(', round(p, 4), ', ', round(sdphat 2, 4), '), n=100', sep='')), bg='gray 90', inset=. 02, lty=c(1, 2), lwd=c(2, 2)) 10

APPENDIX(CONT’D) q R codes for the simulations (N <- length(haswi)) (L <- length(rd)) # prob 1: set. seed(20); n <- 25 ( mystart <- sample(1: L, size=1) ) ( myindex <- rd[mystart+c(1: n)] ) ( mysample <- haswi[myindex] ) # prob 2: ( myphat <- sum(mysample=="Yes")/n ) # prob 3: p <- 0. 5575 ( p - myphat ) # above is for one students. For many students, we have phats set. seed(241); phats <- numeric(nstudents <- 10000) for (t in 1: nstudents){ mystarts <- sample(1: L, size=1) myindexs <- rd[mystarts+c(1: n)] mysamples <- haswi[myindexs] phats[t] <- sum(mysamples=="Yes")/n } phats <- na. omit(phats) # prob 4: ( sdphat <- sqrt(p*(1 -p)/n) ) hist(phats, xlab=expression(hat(p)), freq=F, main ='') vec <- seq(min(phats), max(phats), length=1000); lines(dnorm(vec, p, sdphat)~vec) 11

Thank you. 12