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--- c:ms:2026:lecture_note_week_04 [2026/03/31 16:34] – [Recap] hkimscil
+++ c:ms:2026:lecture_note_week_04 [2026/03/31 22:28] (current) – hkimscil
@@ Line 46: / Line 46: @@
 </code>
-====== Recap ======
-Distribution of Sample Means -- mu = 40, sigma = 4 (hence var = 16) 인 모집단에서 n = n 사이즈의 샘플링을 무한 반복할 때 그 샘플평균들이 모인 집합
-<tabbox rscript01>
-<code>
-rm(list=ls())
-rnorm2 <- function(n,mean,sd){
-  mean+sd*scale(rnorm(n))
-}
-ss <- function(x) {
-  sum((x-mean(x))^2)
-}
-mu <- 40
-sigma <- 4
-iter <- 1000000
-sz <- 16
-se <- sigma/sqrt(16)
-################################
-means <- rnorm2(iter, mu, se)
-hist(means, breaks=50,
-     xlim = c(mu-6*se, mu+6*se),
-     main = paste("sampling distribution"))
-abline(v=mu, col='black', lwd=2)
-lo1 <- mu - se*1
-hi1 <- mu + se*1
-lo2 <- mu - se*2
-hi2 <- mu + se*2
-lo3 <- mu - se*3
-hi3 <- mu + se*3
-abline(v=c(lo1, lo2, lo3, hi1, hi2, hi3),
-       col=c("green","blue", "black"),
-       lwd=2)
-print(c(lo2, hi2))
-m.samp <- 37
-p.val <- pnorm(m.samp, mu, se)*2
-p.val
-z.cal <- (m.samp-mu)/se
-z.cal
-p.val <- pnorm(z.cal)*2
-p.val
-zmeans <- scale(means)
-hist(zmeans, breaks=50,
-     xlim = c(0-10*1, 0+10*1),
-     main=("normalized distribution\nof sample means"))
-abline(v=0, col="black", lwd=2)
-abline(v=z.cal, col='blue', lwd=2)
-abline(v=-z.cal, col="green", lwd=2)
-text(x=-6, y=50000,
-     label=paste("z.cal =", z.cal),
-     pos = 1,
-     col="blue", cex=1)
-text(x=4, y=50000,
-     label=paste(-z.cal),
-     pos=1,
-     col="green", cex=1)
-text(x=-6, y=30000,
-     label=paste("pnorm(z.cal)*2 =", "\n",
-                 round(p.val,3)),
-     pos = 1,
-     col="red", cex=.8)
-hist(zmeans, breaks=50,
-     xlim = c(0-10*1, 0+10*1),
-     main=("normalized distribution\nof sample means"))
-abline(v=0, col="black", lwd=2)
-abline(v=c(-1,-2,-3,1,2,3),
-       col=c("green", "blue", "black"), lwd=2)
-z.cal
-p.val
-#####
-# 위의 아이디어로는 z.cal 점수가
-# +-2 밖에 있는지 보면 된다. 즉,
-# 이는 prob가 0.05보다 작은지
-# 보면 되는 것이다.
-#####
-# +-2 는 정확한 숫자가 아니고
-# qnorm(.05/2) 에 해당하는 숫자
-# 가 정확한 숫자
-two.minus.exact <- qnorm(.05/2)
-two.plus.exact <- qnorm(1-(.05/2))
-c(two.minus.exact, two.plus.exact)
-#####
-# 그러나 R 사용시에는 z 점수로
-# 판단하기 보다는
-# 직접 구하는 prob.로 판단
-pnorm(z.cal)*2
-p.val
-#####
-# 위에서 그룹 간의 차이를
-# standard error로 나누는 것에 주의
-#
-################
-m.samp <- 43
-sd.samp <- 4
-sz <- 16
-samp <- rnorm2(sz, m.samp, sd.samp)
-diff <- m.samp - mu
-se <- sd.samp / sqrt(sz)
-t.cal <- diff/se
-df <- sz-1
-p.val <- pt(t.cal, df=df, lower.tail = F)*2
-t.cal
-df
-p.val
-t.test(samp, mu=mu)
-#####
-#
-m.a <- 5.8
-m.b <- 6.3
-sd.a <- .5
-sd.b <- .5
-sz.a <- 16
-sz.b <- 16
-df.a <- sz.a-1
-df.b <- sz.b-1
-df <- df.a + df.b
-a <- rnorm2(sz.a, m.a, sd.a)
-b <- rnorm2(sz.b, m.b, sd.b)
-diff <- m.a - m.b
-pv <- (ss(a)+ss(b))/(df.a+df.b)
-se <- sqrt(pv/sz.a+pv/sz.b)
-t.cal <- diff / se
-p.val <- pt(t.cal, df=df)*2
-diff
-se
-t.cal
-df
-p.val
-t.test(a,b, var.equal = T)
-diff - se*2
-diff + se*2
-lo <- qt(.05/2,df)
-lo
-hi <- -lo
-diff + se*lo
-diff + se*hi
-</code>
-<tabbox out01>
-<code>
-> rm(list=ls())
-> rnorm2 <- function(n,mean,sd){
-+   mean+sd*scale(rnorm(n))
-+ }
-> ss <- function(x) {
-+   sum((x-mean(x))^2)
-+ }
->
-> mu <- 40
-> sigma <- 4
-> iter <- 1000000
-> sz <- 16
-> se <- sigma/sqrt(16)
-> ################################
-> means <- rnorm2(iter, mu, se)
-> hist(means, breaks=50,
-+      xlim = c(mu-6*se, mu+6*se),
-+      main = paste("sampling distribution"))
-> abline(v=mu, col='black', lwd=2)
-> lo1 <- mu - se*1
-> hi1 <- mu + se*1
-> lo2 <- mu - se*2
-> hi2 <- mu + se*2
-> lo3 <- mu - se*3
-> hi3 <- mu + se*3
->
-> abline(v=c(lo1, lo2, lo3, hi1, hi2, hi3),
-+        col=c("green","blue", "black"),
-+        lwd=2)
->
-> print(c(lo2, hi2))
-[1] 38 42
->
-> m.samp <- 37
-> p.val <- pnorm(m.samp, mu, se)*2
-> p.val
-[1] 0.002699796
-> z.cal <- (m.samp-mu)/se
-> z.cal
-[1] -3
-> p.val <- pnorm(z.cal)*2
-> p.val
-[1] 0.002699796
->
-> zmeans <- scale(means)
-> hist(zmeans, breaks=50,
-+      xlim = c(0-10*1, 0+10*1),
-+      main=("normalized distribution\nof sample means"))
-> abline(v=0, col="black", lwd=2)
-> abline(v=z.cal, col='blue', lwd=2)
-> abline(v=-z.cal, col="green", lwd=2)
-> text(x=-6, y=50000,
-+      label=paste("z.cal =", z.cal),
-+      pos = 1,
-+      col="blue", cex=1)
-> text(x=4, y=50000,
-+      label=paste(-z.cal),
-+      pos=1,
-+      col="green", cex=1)
-> text(x=-6, y=30000,
-+      label=paste("pnorm(z.cal)*2 =", "\n",
-+                  round(p.val,3)),
-+      pos = 1,
-+      col="red", cex=.8)
->
-> hist(zmeans, breaks=50,
-+      xlim = c(0-10*1, 0+10*1),
-+      main=("normalized distribution\nof sample means"))
-> abline(v=0, col="black", lwd=2)
-> abline(v=c(-1,-2,-3,1,2,3),
-+        col=c("green", "blue", "black"), lwd=2)
->
-> z.cal
-[1] -3
-> p.val
-[1] 0.002699796
-> #####
-> # 위의 아이디어로는 z.cal 점수가
-> # +-2 밖에 있는지 보면 된다. 즉,
-> # 이는 prob가 0.05보다 작은지
-> # 보면 되는 것이다.
-> #####
-> # +-2 는 정확한 숫자가 아니고
-> # qnorm(.05/2) 에 해당하는 숫자
-> # 가 정확한 숫자
-> two.minus.exact <- qnorm(.05/2)
-> two.plus.exact <- qnorm(1-(.05/2))
-> c(two.minus.exact, two.plus.exact)
-[1] -1.959964  1.959964
-> #####
-> # 그러나 R 사용시에는 z 점수로
-> # 판단하기 보다는
-> # 직접 구하는 prob.로 판단
-> pnorm(z.cal)*2
-[1] 0.002699796
-> p.val
-[1] 0.002699796
-> #####
-> # 위에서 그룹 간의 차이를
-> # standard error로 나누는 것에 주의
-> #
->
->
-> ################
-> m.samp <- 43
-> sd.samp <- 4
-> sz <- 16
-> samp <- rnorm2(sz, m.samp, sd.samp)
-> diff <- m.samp - mu
-> se <- sd.samp / sqrt(sz)
-> t.cal <- diff/se
-> df <- sz-1
-> p.val <- pt(t.cal, df=df, lower.tail = F)*2
-> t.cal
-[1] 3
-> df
-[1] 15
-> p.val
-[1] 0.008972737
-> t.test(samp, mu=mu)
-	One Sample t-test
-data:  samp
-t = 3, df = 15, p-value = 0.008973
-alternative hypothesis: true mean is not equal to 40
-percent confidence interval:
-.86855 45.13145
-sample estimates:
-mean of x
-
->
-> #####
-> #
-> m.a <- 5.8
-> m.b <- 6.3
-> sd.a <- .5
-> sd.b <- .5
-> sz.a <- 16
-> sz.b <- 16
-> df.a <- sz.a-1
-> df.b <- sz.b-1
-> df <- df.a + df.b
-> a <- rnorm2(sz.a, m.a, sd.a)
-> b <- rnorm2(sz.b, m.b, sd.b)
-> diff <- m.a - m.b
-> pv <- (ss(a)+ss(b))/(df.a+df.b)
-> se <- sqrt(pv/sz.a+pv/sz.b)
-> t.cal <- diff / se
-> p.val <- pt(t.cal, df=df)*2
->
-> diff
-[1] -0.5
-> se
-[1] 0.1767767
-> t.cal
-[1] -2.828427
-> df
-[1] 30
-> p.val
-[1] 0.008257336
-> t.test(a,b, var.equal = T)
-	Two Sample t-test
-data:  a and b
-t = -2.8284, df = 30, p-value = 0.008257
-alternative hypothesis: true difference in means is not equal to 0
-percent confidence interval:
- -0.8610262 -0.1389738
-sample estimates:
-mean of x mean of y
-.8       6.3
-> diff - se*2
-[1] -0.8535534
-> diff + se*2
-[1] -0.1464466
-> lo <- qt(.05/2,df)
-> lo
-[1] -2.042272
-> hi <- -lo
-> diff + se*lo
-[1] -0.8610262
-> diff + se*hi
-[1] -0.1389738
->
-</code>
-</tabbox>
-{{.:pasted:20260331-161501.png}}
-{{.:pasted:20260331-161549.png}}
-{{.:pasted:20260331-161614.png}}
-<tabbox rscript02>
-<code>
-#####
-#
-m.a <- 5.8
-m.b <- 6.3
-sd.a <- .5
-sd.b <- .5
-sz.a <- 16
-sz.b <- 16
-df.a <- sz.a-1
-df.b <- sz.b-1
-df <- df.a + df.b
-a <- rnorm2(sz.a, m.a, sd.a)
-b <- rnorm2(sz.b, m.b, sd.b)
-diff <- m.a - m.b
-pv <- (ss(a)+ss(b))/(df.a+df.b)
-se <- sqrt(pv/sz.a+pv/sz.b)
-t.cal <- diff / se
-p.val <- pt(t.cal, df=df)*2
-diff
-se
-t.cal
-df
-p.val
-t.test(a,b, var.equal = T)
-diff - se*2
-diff + se*2
-lo <- qt(.05/2,df)
-lo
-hi <- -lo
-diff + se*lo
-diff + se*hi
-#####
-# t-test repeated measre
-#####
-m.t1 <- 103
-m.t2 <- 111
-sd.t1 <- 10
-sd.t2 <- 10
-sz <- 160
-t1 <- rnorm2(sz, m.t1, sd.t1)
-t2 <- rnorm2(sz, m.t2, sd.t2)
-t1
-t2
-mdiff <- m.t1-m.t2
-diff <- t1-t2
-sd.diff <- sd(diff)
-se <- sd.diff/sqrt(sz)
-t.cal <- mdiff/se
-p.val <- pt(t.cal, df=sz-1)*2
-t.cal
-sz-1
-p.val
-t.test(t1,t2, paired=T)
-two <- qt(.05/2, df=sz-1)
-two
-lo <- se*two
-hi <- -lo
-c(lo, hi)
-c(mdiff+lo, mdiff+hi)
-</code>
-<tabbox rout02>
-<code>
->
->
->
->
-> #####
-> #
-> m.a <- 5.8
-> m.b <- 6.3
-> sd.a <- .5
-> sd.b <- .5
-> sz.a <- 16
-> sz.b <- 16
-> df.a <- sz.a-1
-> df.b <- sz.b-1
-> df <- df.a + df.b
-> a <- rnorm2(sz.a, m.a, sd.a)
-> b <- rnorm2(sz.b, m.b, sd.b)
-> diff <- m.a - m.b
-> pv <- (ss(a)+ss(b))/(df.a+df.b)
-> se <- sqrt(pv/sz.a+pv/sz.b)
-> t.cal <- diff / se
-> p.val <- pt(t.cal, df=df)*2
->
-> diff
-[1] -0.5
-> se
-[1] 0.1767767
-> t.cal
-[1] -2.828427
-> df
-[1] 30
-> p.val
-[1] 0.008257336
-> t.test(a,b, var.equal = T)
-	Two Sample t-test
-data:  a and b
-t = -2.8284, df = 30, p-value = 0.008257
-alternative hypothesis: true difference in means is not equal to 0
-percent confidence interval:
- -0.8610262 -0.1389738
-sample estimates:
-mean of x mean of y
-.8       6.3
-> diff - se*2
-[1] -0.8535534
-> diff + se*2
-[1] -0.1464466
-> lo <- qt(.05/2,df)
-> lo
-[1] -2.042272
-> hi <- -lo
-> diff + se*lo
-[1] -0.8610262
-> diff + se*hi
-[1] -0.1389738
->
-> #####
-> # t-test repeated measre
-> #####
-> m.t1 <- 103
-> m.t2 <- 111
-> sd.t1 <- 10
-> sd.t2 <- 10
-> sz <- 160
-> t1 <- rnorm2(sz, m.t1, sd.t1)
-> t2 <- rnorm2(sz, m.t2, sd.t2)
-> t1
-            [,1]
-  [1,] 111.36104
-  [2,]  88.01731
-  [3,] 101.88320
-  [4,] 103.10436
-  [5,]  96.38094
-  [6,] 110.28539
-  [7,]  98.80045
-  [8,]  86.31780
-  [9,] 128.73687
- [10,] 102.23723
- [11,]  94.90932
- [12,] 110.74820
- [13,] 108.90433
- [14,]  91.11044
- [15,]  83.69413
- [16,]  84.12565
- [17,] 110.63493
- [18,]  77.88247
- [19,] 114.62381
- [20,] 109.08372
- [21,] 105.43251
- [22,]  93.28446
- [23,] 112.45340
- [24,] 108.82774
- [25,] 101.87046
- [26,]  99.06745
- [27,] 105.41497
- [28,]  89.35983
- [29,] 128.33953
- [30,] 101.82933
- [31,]  84.43396
- [32,] 107.31985
- [33,]  94.50306
- [34,]  98.51383
- [35,] 100.37196
- [36,] 104.39854
- [37,]  99.00553
- [38,] 100.39100
- [39,]  98.94237
- [40,]  94.04721
- [41,]  91.55691
- [42,]  77.02969
- [43,] 100.65928
- [44,]  99.50989
- [45,] 113.40564
- [46,]  91.27212
- [47,]  96.54430
- [48,] 103.67181
- [49,]  91.91200
- [50,]  95.86468
- [51,]  97.73431
- [52,] 105.95878
- [53,]  99.40692
- [54,] 114.89231
- [55,] 110.23953
- [56,] 110.65776
- [57,]  95.35294
- [58,] 114.74190
- [59,] 107.10249
- [60,]  97.93327
- [61,] 114.29149
- [62,] 106.77413
- [63,]  89.85116
- [64,] 100.92937
- [65,] 110.57659
- [66,] 118.43433
- [67,]  97.26787
- [68,] 112.06303
- [69,] 101.08834
- [70,] 112.54527
- [71,] 103.74242
- [72,] 107.31976
- [73,] 114.14557
- [74,]  96.41347
- [75,]  96.73140
- [76,]  92.48801
- [77,]  93.13216
- [78,]  93.39353
- [79,] 106.83687
- [80,]  95.43550
- [81,]  99.92717
- [82,] 105.47433
- [83,]  88.13565
- [84,] 104.37033
- [85,]  96.23481
- [86,] 105.73652
- [87,]  99.62358
- [88,] 112.79561
- [89,] 111.78083
- [90,] 114.73846
- [91,]  98.61353
- [92,] 121.41442
- [93,] 104.81865
- [94,] 100.92946
- [95,] 107.41369
- [96,]  98.22645
- [97,] 104.94036
- [98,]  93.38986
- [99,] 107.18154
-[100,] 108.80844
-[101,] 117.97939
-[102,] 103.40657
-[103,]  99.54187
-[104,]  98.22691
-[105,]  99.13327
-[106,]  93.54839
-[107,]  99.47141
-[108,]  72.82718
-[109,] 120.41493
-[110,] 106.81977
-[111,] 104.10554
-[112,]  92.11256
-[113,] 117.84020
-[114,] 106.80209
-[115,] 123.11219
-[116,] 112.60503
-[117,] 113.01015
-[118,]  95.06184
-[119,]  97.10124
-[120,]  88.02648
-[121,] 103.98118
-[122,] 112.38688
-[123,] 100.76566
-[124,] 104.56130
-[125,] 110.20566
-[126,] 108.55945
-[127,] 101.47467
-[128,] 100.21853
-[129,] 103.10659
-[130,]  95.19338
-[131,]  98.03036
-[132,] 107.44486
-[133,] 100.49136
-[134,] 105.64403
-[135,] 103.33323
-[136,] 111.37567
-[137,]  88.13074
-[138,] 106.90384
-[139,] 100.01857
-[140,] 110.50553
-[141,] 124.36441
-[142,] 106.98552
-[143,] 115.77759
-[144,] 101.10420
-[145,] 105.26656
-[146,]  93.98217
-[147,] 120.60988
-[148,]  94.68497
-[149,] 127.71822
-[150,] 128.63994
-[151,] 106.18538
-[152,]  92.98331
-[153,]  99.14643
-[154,] 110.37932
-[155,] 104.60248
-[156,] 106.81372
-[157,]  94.45348
-[158,] 113.53202
-[159,] 107.81640
-[160,]  87.36641
-attr(,"scaled:center")
-[1] 0.02797819
-attr(,"scaled:scale")
-[1] 1.028345
-> t2
-            [,1]
-  [1,] 104.52214
-  [2,] 106.01539
-  [3,] 118.86196
-  [4,] 110.72811
-  [5,] 121.76890
-  [6,] 104.08372
-  [7,] 104.79700
-  [8,] 130.54567
-  [9,] 104.51772
- [10,]  98.46445
- [11,] 102.79140
- [12,] 120.40580
- [13,] 112.91734
- [14,] 113.77920
- [15,] 114.20003
- [16,] 107.49174
- [17,] 101.19277
- [18,] 115.36843
- [19,] 119.91569
- [20,] 106.06605
- [21,] 123.11635
- [22,]  93.79225
- [23,]  93.48746
- [24,] 117.74609
- [25,] 109.58166
- [26,] 134.83143
- [27,]  98.45053
- [28,] 106.25705
- [29,] 100.76346
- [30,] 117.52330
- [31,]  99.08305
- [32,] 120.38723
- [33,] 116.10505
- [34,] 120.00785
- [35,] 116.23227
- [36,] 116.00613
- [37,] 124.99957
- [38,] 115.16024
- [39,] 114.95141
- [40,]  98.03156
- [41,] 109.35921
- [42,] 108.94960
- [43,] 106.56490
- [44,] 116.28102
- [45,] 116.59853
- [46,] 108.34954
- [47,] 113.88005
- [48,]  89.61658
- [49,] 106.68461
- [50,] 124.51694
- [51,] 124.60305
- [52,] 103.93134
- [53,] 118.46683
- [54,] 118.84622
- [55,] 118.55730
- [56,] 120.96029
- [57,] 120.91002
- [58,]  93.65926
- [59,] 118.82763
- [60,] 113.24234
- [61,] 113.75956
- [62,] 111.12494
- [63,] 107.98393
- [64,] 118.47903
- [65,] 108.81494
- [66,] 122.69894
- [67,] 108.42655
- [68,] 130.67077
- [69,]  97.24069
- [70,] 110.17917
- [71,] 103.99463
- [72,] 125.31486
- [73,]  91.26600
- [74,] 105.84776
- [75,] 123.22794
- [76,] 110.03864
- [77,] 115.89615
- [78,] 112.08779
- [79,] 112.91045
- [80,] 120.60075
- [81,] 105.78417
- [82,]  92.46600
- [83,]  88.53759
- [84,] 127.18477
- [85,] 122.35360
- [86,] 123.61826
- [87,] 110.69036
- [88,] 110.69824
- [89,] 107.37120
- [90,] 107.72845
- [91,] 112.04867
- [92,]  96.05635
- [93,] 108.67360
- [94,] 118.85047
- [95,] 103.94559
- [96,] 110.11686
- [97,] 122.59443
- [98,] 110.32224
- [99,] 105.05468
-[100,] 119.83378
-[101,] 116.87027
-[102,] 113.08225
-[103,] 109.16396
-[104,] 108.50073
-[105,] 105.41310
-[106,]  95.32322
-[107,] 105.61400
-[108,] 129.13587
-[109,]  86.96069
-[110,] 115.11906
-[111,] 106.76034
-[112,] 102.69395
-[113,]  97.25614
-[114,] 105.64480
-[115,] 111.23453
-[116,] 117.35671
-[117,] 120.65920
-[118,] 108.53758
-[119,] 109.18312
-[120,] 112.51380
-[121,] 119.76297
-[122,] 132.25435
-[123,] 118.96046
-[124,] 108.72113
-[125,] 124.70416
-[126,]  91.89676
-[127,] 117.43380
-[128,] 116.62698
-[129,]  94.52859
-[130,] 106.74281
-[131,] 109.20631
-[132,] 112.79079
-[133,] 113.50772
-[134,]  91.10812
-[135,]  77.92339
-[136,] 118.20013
-[137,] 106.33092
-[138,] 112.72908
-[139,] 110.12456
-[140,] 114.61151
-[141,] 123.38285
-[142,] 119.22545
-[143,] 116.54886
-[144,] 111.95272
-[145,] 112.92325
-[146,] 104.37119
-[147,] 110.30857
-[148,] 103.80554
-[149,] 131.22599
-[150,] 107.36467
-[151,] 110.81009
-[152,]  95.49339
-[153,] 107.73582
-[154,] 105.47526
-[155,] 119.48971
-[156,] 114.18188
-[157,]  93.91213
-[158,] 107.63677
-[159,] 125.62210
-[160,] 101.10034
-attr(,"scaled:center")
-[1] -0.03441822
-attr(,"scaled:scale")
-[1] 0.9901942
-> mdiff <- m.t1-m.t2
-> diff <- t1-t2
-> sd.diff <- sd(diff)
-> se <- sd.diff/sqrt(sz)
-> t.cal <- mdiff/se
-> p.val <- pt(t.cal, df=sz-1)*2
-> t.cal
-[1] -7.249999
-> sz-1
-[1] 159
-> p.val
-[1] 1.713273e-11
-> t.test(t1,t2, paired=T)
-	Paired t-test
-data:  t1 and t2
-t = -7.25, df = 159, p-value = 1.713e-11
-alternative hypothesis: true mean difference is not equal to 0
-percent confidence interval:
- -10.179307  -5.820693
-sample estimates:
-mean difference
-             -8
-> two <- qt(.05/2, df=sz-1)
-> two
-[1] -1.974996
-> lo <- se*two
-> hi <- -lo
-> c(lo, hi)
-[1] -2.179307  2.179307
-> c(mdiff+lo, mdiff+hi)
-[1] -10.179307  -5.820693
->
-</code>
-</tabbox>