STT 200 – L ECTURE 5, SECTION 23,24 R ECITATION 11 (3/26/2013) TA: Zhen Zhang Office hour: (C500 WH) 3-4 PM Tuesday (office tel.: ) Help-room: (A102 WH) 9:00AM-1:00PM, Monday Class meet on Tuesday: 12:40 – 1:30PM A224 WH, Section 23 1:50 – 2:40PM A234 WH, Section 24 1

M AIN G OALS 2

D ATA 3  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","Yes","Yes","No","Yes","No","Yes","No","Yes","Yes","No","Yes","Yes","Yes", "No","Yes","No","No","Yes","No","No","No","No","No","Yes","Yes","Yes","No","Yes","Yes","Yes ","No","No","No","Yes","Yes","No","Yes","Yes","Yes","Yes","No","No","No","No","No","No","Ye s","Yes","No","Yes","No","No","Yes","No","No","No","No","Yes","Yes","Yes","Yes","No","Yes", "Yes","Yes","No","No","Yes","No","No","Yes","No","No","No","No","No","Yes","Yes","No","Yes","No","Yes","Yes","Yes","Yes","No","Yes","Yes","Yes","No","Yes","No","Yes","Yes","No","No", "No","No","Yes","No","Yes","No","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","No", "No","Yes","No","Yes","Yes","Yes","No","No","Yes","No","Yes","Yes","Yes","No","Yes","No","N o","No","Yes","No","Yes","Yes","No","Yes","Yes","Yes","No","Yes","No","Yes","No","No","No", "Yes","No","Yes","Yes","Yes","Yes","No","Yes","Yes","Yes","Yes","Yes","No","Yes","Yes","Yes ","No","Yes","No","No","Yes","Yes","No","Yes","Yes","Yes","No","No","Yes","Yes","Yes","Yes","No","Yes","Yes","Yes","Yes","Yes","No","Yes","Yes","Yes","No","Yes","Yes","Yes","Yes","No ","Yes","No","No","Yes","No","Yes","Yes","No","Yes","Yes","No","No","Yes","No","Yes","Yes", "Yes","No","Yes","Yes","No","No","Yes","No","Yes","Yes","No","Yes","Yes","Yes","No","Yes"," Yes","No","No","No","Yes","Yes","Yes","No","Yes","Yes","Yes","Yes","Yes","No","Yes","Yes"," No","Yes","No","No","Yes","Yes","No","Yes","Yes","No","Yes","Yes","No","Yes","No","No","Yes ","Yes","No","Yes","No","Yes","No","Yes","No","Yes","No","No","Yes","No","Yes","Yes","No"," Yes","Yes","Yes","No","Yes","No","Yes","No","Yes","Yes","No","No","Yes","Yes","Yes","No","Y es","No","No","Yes","No","Yes","Yes","Yes","No","No","Yes","No","Yes","No","Yes","Yes","Yes ","No","No","No","Yes","No","No","No","Yes","Yes","Yes","No","Yes","No","Yes","No","No","Ye s","Yes","Yes","No","Yes","No","Yes","No","No","Yes","Yes","Yes","No","No","No","No","Yes", "No","No","No","No","No","Yes","No","Yes","Yes","No","Yes","Yes","Yes","Yes","No","No","No","Yes","No","No","Yes","Yes","No","Yes","Yes","Yes","No","Yes","No","No","No","No","Yes","N o","No","No","Yes","Yes","Yes","Yes","Yes","No","Yes","Yes","No","Yes","Yes","No","Yes","No ","Yes","No","No","No","Yes","No","No","Yes","No")

D ATA 4  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, 117,277,93,399,226,8,231,386,136,75,38,56,37,267,381,63,52,231,287,94,50,77,179,337,387,318,112,219,17,356,77,183,259,258,141,198,30,36,61,306,65,330,161,348,19,20,61,275,365,241,115,4,338,205,108,241,190,374,323,243,146,318,217,375,267,44,373,185,341,283,200,1 78,266,390,232,263,386,36,270,50,315,83,90,281,260,41,305,136,116,185,25,338,4,367,296, 183,103,290,208,170,143,158,198,132,155,144,26,104,281,150,240,68,67,339,389,345,141,26 8,349,99,147,65,170,375,317,251,185,278,80,250,4,378,175,130,359,319,400,59,166,147,130,107,123,304,234,41,20,165,96,115,272,149,142,75,262,235,106,107,354,362,2,81,89,309,37 1,10,282,202,203,156,386,130,252,26,387,143,237,183,328,306,27,187,310,321,183,109,198, 200,281,70,394,378,203,42,34,318,156,255,354,53,196,20,382,97,292,188,179,69,151,14,348,311,389,298,399,104,300,243,163,316,328,65,167,200,301,305,27,176,69,301,188,192,242,3 50,92,86,42,373,195,118,64,289,329,131,156,252,169,299,191,302,19,83,220,326,229,285,26 7,351,333,101,128,146,307,304,245,264,149,163,353,276,296,243,8,127,31,210,263,33,384,1 76,125,275,76,45,60,59,143,324,281,376,298,54,62,170,295,293,27,183,126,375,21,294,242, 364,145,138,52,267,26,308,391,352,78,98,211,174,277,176,74,295,64,315,171,135,159,111,7 9,348,88,23,348,111,188,16,152,212,104,349,14,272,209,73,238,146,50,113,103,204,389,158,260,344,207,329,184,250,38,231,292,300,34,170,343,233,275,14,15,244,104,96,234,297,113,270,369,202,37,310,294,64,183,253,299,287,225,166,260,125,198,2,180,219,117,358,191,30 1,310,254,230,296,2,134,67,186,265,161,130,257,166,339,33,332,137,61,340,16,212,209,42, 315,8,269,68,389,316,355,62,51,64,388,260,319,244,116,265,169,153,147,170,59,329,261,38 4,272,367,177,217,278,266,307,182,225,80,264,342,280,350,366,280,156,323,208,110,37,266,260,59,33,314,80,185,185,87,228,246,61,369,60,119,179,326,223,128,62,98,130,283,328,22 5,398,3,138,140,84,381,234,131,364,294,59,343,126,93,14,204,50,35,161,15,142,275,72,254,194,309,115,344,378,267,23,111,168,334,92,213,1,181,246,336,52,82,4,115,286,3,87,121,8 4,281,181,58,372,232,30,279,258,154,37,6,113,125,317,123,198,25,388,268,106)

P ROBLEMS 5

S IMULATION 6

P ROBLEMS 7

8

9

A PPENDIX 10  R codes for the problems. # prob 4: n <- 25; p < ( sdphat <- sqrt(p*(1-p)/n) ) # prob 5: ( pnorm(p+0.1, p, sdphat) - pnorm(p-0.1, p, sdphat) ) # prob 6: n2 <- 100 ( sdphat2 <- sqrt(p*(1-p)/n2) ) ( pnorm(p+0.1, p, sdphat2) - pnorm(p-0.1, p, sdphat2) ) # comparison of n=25 and n=100 vec <- seq(0.01,0.99, length=1000) par(yaxt='n',mar=c(4,.3,.3,.3)) plot(dnorm(vec, p, sdphat2)~vec, type='n', ylab=' ',xlab=expression(hat(p))) grid(col='gray80') lines(dnorm(vec, p, sdphat)~vec, lty=1, lwd=2) lines(dnorm(vec, p, sdphat2)~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(sdphat2,4),'), n=100',sep='')), bg='gray90', inset=.02, lty=c(1,2), lwd=c(2,2))

A PPENDIX ( CONT ’ D ) 11  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 < ( p - myphat ) # above is for one students. For many students, we have phats set.seed(241); phats <- numeric(nstudents < ) 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)

Thank you. 12