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--- t-test_summary [2026/04/12 06:32] – [rs.hypothesis.testing] hkimscil
+++ t-test_summary [2026/04/16 01:27] (current) – hkimscil
@@ Line 29: / Line 29: @@
 ################################
-sz <- 10
+sz <- 16
 iter <- 100000
-# n = 10 일 때의 p1에 대한 sampling dist 은 아래 시뮬레이션으로
+# n = 16 일 때의 p1에 대한 sampling dist 은 아래 시뮬레이션으로
 # 구해볼 수 있다.
 means <- rep(NA, iter)
@@ Line 50: / Line 50: @@
 sd.means
-# 위의 시뮬레이션으로 구한 sampling dist 대신에
+# 위 집합을 표준화하게 되면 그 집합의 평균과
-# 정확한 mean과 se 값을 갖는 집합 sdc를 만든다
+# 표준편차는 0, 1이 (분산도 1) 된다.
-sdc <- rnorm2(iter, m.means, sd.means)
+m.zmeans <- 0
-mean(sdc)
+ms.zmeans <- 1
-var(sdc)
+sd.zmeans <- 1
-sd(sdc)
-zsdc <- scale(sdc)
+sd.means
-m.zsdc <- mean(zsdc)
+# p2에서 구하는 샘플링 디스트리뷰션의 표준점수를
-ms.zsdc <- var(zsdc)
+# p1의 표준점수에 (0, 1) 비교해서 배치하자면
-sd.zsdc <- sd(zsdc)
+z.p2 <- c((mean(p2)-mean(p1))/sd.means) # 표준점수 평균
-m.zsdc
+sd.means <- c(sqrt(var(p1)/sz)) # 표준편차
-ms.zsdc
-sd.zsdc
-se2 <- c(sqrt(var(p2)/sz))
-z.p2 <- c((mean(p2)-mean(p1))/se2)
-sdc2 <- rnorm2(iter, mean(p2), se2)
-zsdc2 <- scale(sdc2)+z.p2
-mean(zsdc2)
-sd(zsdc2)
 curve(dnorm(x), from = -4, to = z.p2+4,
-      main = "normalized distribution of sample means from p1 and p2",
+      main = "normalized distribution of sample \n means from p1 and p2 (n=10)",
-      ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
+      ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
-curve(dnorm(x-(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
+curve(dnorm(x-c(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
-      main = "Distribution Curve",
+      main = "",
-      ylab = "Density", xlab = "t-value", col = "blue", lwd = 2, lty=2)
+      ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
-abline(v=mean(zsdc), col='black', lwd=2)
+abline(v=0, col='black', lwd=2)
-abline(v=mean(zsdc2), col='blue', lwd=2)
+abline(v=z.p2, col='blue', lwd=2)
-mean(zsdc2)
+text(x=0, y=.1, label=paste(round(0, 4)), pos=4)
-text(x=mean(zsdc), y=.1, label=paste(round(mean(zsdc),4)), pos=4)
+text(x=z.p2, y=.1, label=paste(round(z.p2, 4)), pos=4)
-text(x=mean(zsdc2), y=.1, label=paste(round(mean(zsdc2),4)), pos=4)
-#
+#######################################
+#######################################
 lo1 <- qnorm(.32/2)
 hi1 <- -lo1
@@ Line 106: / Line 96: @@
 text(x=hi3, y=.1, label=paste(round(hi3,3), "(3)", "\n","99%"), pos=4)
-mean.of.sample.a <- mean(sdc)+ 1.5*sd(sdc)
+mean.of.sample.a <- m.means+ 1.5*sd.means
 mean.of.sample.a
-diff <- (mean.of.sample.a - mean(sdc))
+diff <- (mean.of.sample.a - m.means)
 se.z <- sd(p1)/sqrt(sz)
 diff
@@ Line 130: / Line 120: @@
 # 새로운 UI로 게임을 하도록 한 후
-# UI점수를 10명에게 구했다고 가정하고
+# UI점수를 sz 명에게 구했다고 가정하고
 # 새로운 UI점수가 기존의 p1 paramter와
 # 다른지 테스트 해보라
@@ Line 138: / Line 128: @@
 # 하면 샘플의 평균과 p1의 평균은 다르다고 판단될 것이다.
 # 아래는 그럼에도 불구하고 실패하는 경우이다.
-set.seed(111)
+set.seed(110)
 smp <- sample(p2, sz, replace=T)
 m.smp <- mean(smp)
@@ Line 150: / Line 140: @@
 curve(dnorm(x), from = -4, to = z.p2+4,
-      main = "normalized distribution of sample means \n testing with a sample from p2 (failed)",
+      main = "normalized distribution of sample means
+      testing with a sample from p2 (failed)",
       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 abline(v=0, col="black", lwd=2)
@@ Line 174: / Line 165: @@
 z.test(smp, mean(p1), sd(p1))
-z.p2 <- (mean(p2)-mean(p1))/se2
+z.p2 <- (mean(p2)-mean(p1))/se.z
 z.p2
 curve(dnorm(x), from = -5, to = z.p2+5,
@@ Line 196: / Line 187: @@
 # type i and type ii error
-z.p2 <- (mean(p2)-mean(p1))/se2
+two <- qnorm(.05/2)
-z.p2
+two
 curve(dnorm(x), from = -4.7, to = z.p2+4,
       main = "Distribution Curve",
@@ Line 205: / Line 197: @@
       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
 abline(v=0, col='black', lwd=2)
-z.cal1
-z.cal2
-two <- qnorm(.05/2)
-two
 abline(v=c(two, -two), col='black', lwd=2)
 abline(v=c(-z.cal1, z.cal1), col='red', lwd=2)
@@ Line 263: / Line 251: @@
 print(c(t.cal, df.smp, prob))
 print(c(m.smp+lo2*se.z, m.smp+hi2*se.z))
-cat("t =", t.cal, ", df =", round(df.smp,0), ", p-value =", prob,
+cat(" t =", t.cal, ", df =", round(df.smp,0), ", p-value =", prob,
 "\n", "95% confidence interval =", m.smp+lo2*se.z, m.smp+hi2*se.z)
 t.test(smp, mu=mean(p1))
@@ Line 323: / Line 311: @@
 t.test(group.a, group.b, var.equal = T)
 t.cal
-# t.cal=diff/se
-t.cal * se.s
-diff
-diff+lo2*se.s
-diff+hi2*se.s
-(t.cal+lo2)*se.s
-(t.cal+hi2)*se.s
 ######################
@@ Line 401: / Line 382: @@
 >
 > ################################
+> set.seed(1001)
 > N.p <- 1000000
 > m.p <- 100
@@ Line 408: / Line 390: @@
 > p2 <- rnorm2(N.p, m.p+6, sd.p)
 >
-</code>
-<code>
 > ################################
-> sz <- 10
+> sz <- 16
 > iter <- 100000
-> # n = 10 일 때의 p1에 대한 sampling dist 은 아래 시뮬레이션으로
+> # n = 16 일 때의 p1에 대한 sampling dist 은 아래 시뮬레이션으로
 > # 구해볼 수 있다.
 > means <- rep(NA, iter)
 > for (i in 1:iter) {
-+   # means <- append(means, mean(sample(p1, s.size, replace = T)))
 +   s1 <- sample(p1, sz, replace = T)
 +   means[i] <- mean(s1)
 + }
 > mean(means)
-[1] 99.98864
+[1] 99.997
 > var(means)
-[1] 9.964541
+[1] 6.215719
 > sd(means)
-[1] 3.156666
+[1] 2.493134
 >
 > # CLT에 의하면 위이 값은
@@ Line 436: / Line 414: @@
 [1] 100
 > ms.means
-[1] 10
+[1] 6.25
 > sd.means
-[1] 3.162278
+[1] 2.5
 >
-> # 위의 시뮬레이션으로 구한 sampling dist 대신에
+> # 위 집합을 표준화하게 되면 그 집합의 평균과
-> # 정확한 mean과 se 값을 갖는 집합 sdc를 만든다
+> # 표준편차는 0, 1이 (분산도 1) 된다.
-> sdc <- rnorm2(iter, m.means, sd.means)
+> m.zmeans <- 0
-> mean(sdc)
+> ms.zmeans <- 1
-[1] 100
+> sd.zmeans <- 1
-> var(sdc)
-     [,1]
-[1,]   10
-> sd(sdc)
-[1] 3.162278
 >
-> zsdc <- scale(sdc)
+> sd.means
-> m.zsdc <- mean(zsdc)
+[1] 2.5
-> ms.zsdc <- var(zsdc)
+> # p2에서 구하는 샘플링 디스트리뷰션의 표준점수를
-> sd.zsdc <- sd(zsdc)
+> # p1의 표준점수에 (0, 1) 비교해서 배치하자면
-> m.zsdc
+> z.p2 <- c((mean(p2)-mean(p1))/sd.means) # 표준점수 평균
-[1] -5.276736e-18
+> sd.means <- c(sqrt(var(p1)/sz)) # 표준편차
-> ms.zsdc
-     [,1]
-[1,]    1
-> sd.zsdc
-[1] 1
 >
-> se2 <- c(sqrt(var(p2)/sz))
-> z.p2 <- c((mean(p2)-mean(p1))/se2)
-> sdc2 <- rnorm2(iter, mean(p2), se2)
-> zsdc2 <- scale(sdc2)+z.p2
-> mean(zsdc2)
-[1] 1.897367
-> sd(zsdc2)
-[1] 1
->
-> col1 <- rgb(0, 1, 1, alpha = 0.1)
-> col2 <- rgb(1, 1, 1, alpha = 0.1)
 > curve(dnorm(x), from = -4, to = z.p2+4,
-+       main = "distribution Curve",
++       main = "normalized distribution of sample \n means from p1 and p2 (n=10)",
-+       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
++       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
-> curve(dnorm(x-(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
+> curve(dnorm(x-c(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
-+       main = "Distribution Curve",
++       main = "",
-+       ylab = "Density", xlab = "t-value", col = "blue", lwd = 2, lty=2)
++       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
-> abline(v=mean(zsdc), col='black', lwd=2)
+> abline(v=0, col='black', lwd=2)
-> abline(v=mean(zsdc2), col='blue', lwd=2)
+> abline(v=z.p2, col='blue', lwd=2)
-> mean(zsdc2)
+> text(x=0, y=.1, label=paste(round(0, 4)), pos=4)
-[1] 1.897367
+> text(x=z.p2, y=.1, label=paste(round(z.p2, 4)), pos=4)
-> text(x=mean(zsdc), y=.1, label=paste(round(mean(zsdc),4)), pos=4)
-> text(x=mean(zsdc2), y=.1, label=paste(round(mean(zsdc2),4)), pos=4)
 >
-> #
+</code>
+{{pasted:20260412-232126.png}}
+<code>
+> #######################################
+> #######################################
 > lo1 <- qnorm(.32/2)
 > hi1 <- -lo1
@@ Line 501: / Line 461: @@
 [1] -2.575829  2.575829
 >
-> curve(dnorm(x), from = -4, to = z.p2+4,
+> curve(dnorm(x), from = -4, to = 2+4,
-+       main = "distribution Curve",
++       main = "normalized distribution of sample means from p1",
-+       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
++       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2)
 > abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
 +        col=c("red","red", "blue", "blue", "orange", "orange"),
 +        lwd=2)
+> text(x=hi1, y=.2, label=paste(round(hi1,3), "(1)", "\n","86%"), pos=4)
+> text(x=hi2, y=.15, label=paste(round(hi2,3),"(2)", "\n","95%"), pos=4)
+> text(x=hi3, y=.1, label=paste(round(hi3,3), "(3)", "\n","99%"), pos=4)
 >
-> mean.of.sample.a <- mean(sdc)+ 1.5*sd(sdc)
+</code>
+{{pasted:20260412-063531.png}}
+<code>
+> mean.of.sample.a <- m.means+ 1.5*sd.means
 > mean.of.sample.a
-[1] 104.7434
+[1] 103.75
-> diff <- (mean.of.sample.a - mean(sdc))
+> diff <- (mean.of.sample.a - m.means)
 > se.z <- sd(p1)/sqrt(sz)
 > diff
-[1] 4.743416
+[1] 3.75
 > se.z
-[1] 3.162278
+[1] 2.5
 > z.score  <- diff / se.z
 > z.score
@@ Line 526: / Line 493: @@
 >
 > curve(dnorm(x), from = -4, to = z.p2+4,
-+       main = "distribution curve",
++       main = "normalized distribution of sample means from p1 with z score 1.5",
 +       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2)
@@ Line 537: / Line 504: @@
 +      pos=4, col='red')
 >
+</code>
+{{pasted:20260412-063608.png}}
+<code>
 > # 새로운 UI로 게임을 하도록 한 후
 > # UI점수를 10명에게 구했다고 가정하고
@@ Line 546: / Line 517: @@
 > # 하면 샘플의 평균과 p1의 평균은 다르다고 판단될 것이다.
 > # 아래는 그럼에도 불구하고 실패하는 경우이다.
-> set.seed(5)
+> set.seed(110)
 > smp <- sample(p2, sz, replace=T)
 > m.smp <- mean(smp)
 > m.smp
-[1] 104.5279
+[1] 104.5958
 > diff <- m.smp - mean(p1)
 > se.z <- sqrt(var(p1)/sz)
@@ Line 556: / Line 527: @@
 > prob1 <- pnorm(abs(z.cal1), lower.tail = F)*2
 > print(c(z.cal1, sz, prob1))
-[1]  1.4318575 10.0000000  0.1521846
+[1]  2.906626913 40.000000000  0.003653487
 > z.test(smp, mean(p1), sd(p1))
- z value: 1.43186
+ z value: 2.90663
- p value: 0.1521846
+ p value: 0.00365349
- diff:    104.5279 - 100 = 4.527931
+ diff:    104.5958 - 100 = 4.595781
- se:      3.162278
+ se:      1.581139
-% CI:  93.80205 106.198>
+% CI:  96.90102 103.099>
 > curve(dnorm(x), from = -4, to = z.p2+4,
-+       main = "distribution curve",
++       main = "normalized distribution of sample means
++       testing with a sample from p2 (failed)",
 +       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2)
@@ Line 576: / Line 548: @@
 >
 >
+</code>
+{{pasted:20260412-233253.png}}
+<code>
 > # 같은 방법으로 했는데 성공한 경우
-> set.seed(111)
+> set.seed(211)
 > smp <- sample(p2,sz,replace=T)
 > m.smp <- mean(smp)
 > m.smp
-[1] 110.0083
+[1] 107.6795
 > diff <- m.smp - mean(p1)
 > se.z <- sqrt(var(p1)/sz)
@@ Line 587: / Line 563: @@
 > prob2 <- pnorm(abs(z.cal2), lower.tail = F)*2
 > print(c(z.cal2, sz, prob2))
-[1]  3.16488922 10.00000000  0.00155142
+[1] 4.856940e+00 4.000000e+01 1.192138e-06
 > z.test(smp, mean(p1), sd(p1))
- z value: 3.16489
+ z value: 4.85694
- p value: 0.00155142
+ p value: 1.19e-06
- diff:    110.0083 - 100 = 10.00826
+ diff:    107.6795 - 100 = 7.679496
- se:      3.162278
+ se:      1.581139
-% CI:  93.80205 106.198>
+% CI:  96.90102 103.099>
-> z.p2 <- (mean(p2)-mean(p1))/se2
+> z.p2 <- (mean(p2)-mean(p1))/se.z
 > z.p2
-[1] 1.897367
+         [,1]
-> curve(dnorm(x), from = -4.7, to = z.p2+4,
+[1,] 3.794733
-+       main = "Distribution Curve",
+> curve(dnorm(x), from = -5, to = z.p2+5,
++       main = "normalized distribution of sample means \n testing with a sample from p2 (succeeded)",
 +       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
-> curve(dnorm(x-(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
-+       main = "Distribution Curve",
-+       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
 > abline(v=0, col='black', lwd=2)
 > z.cal1
          [,1]
-[1,] 1.431858
+[1,] 2.906627
 > z.cal2
-         [,1]
+        [,1]
-[1,] 3.164889
+[1,] 4.85694
 > two <- qnorm(.05/2)
 > two
@@ Line 624: / Line 598: @@
 >
 >
+</code>
+{{pasted:20260412-233208.png}}
+<code>
 > # type i and type ii error
-> z.p2 <- (mean(p2)-mean(p1))/se2
+> two <- qnorm(.05/2)
-> z.p2
+> two
-[1] 1.897367
+[1] -1.959964
+>
 > curve(dnorm(x), from = -4.7, to = z.p2+4,
 +       main = "Distribution Curve",
 +       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
-> curve(dnorm(x-(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
+> curve(dnorm(x-c(z.p2)), from = z.p2-3, to = z.p2+3, add = T,
 +       main = "Distribution Curve",
 +       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
 > abline(v=0, col='black', lwd=2)
-> z.cal1
-         [,1]
-[1,] 1.431858
-> z.cal2
-         [,1]
-[1,] 3.164889
-> two <- qnorm(.05/2)
-> two
-[1] -1.959964
 > abline(v=c(two, -two), col='black', lwd=2)
 > abline(v=c(-z.cal1, z.cal1), col='red', lwd=2)
@@ Line 664: / Line 634: @@
 >
 >
+</code>
+{{pasted:20260412-233411.png}}
+<code>
 > ############################
 > # one sample t-test
 > ############################
+> set.seed(99)
 > sz <- 20
 > smp <- sample(p2, sz, replace = T)
@@ Line 676: / Line 651: @@
 > prob <- pt(t.cal, df.smp, lower.tail = F)*2
 > se.z
-[1] 2.58698
+[1] 1.809134
 > qt(.05/2, df.smp)
 [1] -2.093024
@@ Line 682: / Line 657: @@
 > hi2 <- -lo2
 > m.smp+lo2*se.z
-[1] 99.55202
+[1] 102.5239
 >
-> curve(dt(x, df.smp), from = -4, to = 6,
+> curve(dt(x, df.smp), from = -6, to = 7,
-+       main = "t Distribution Curve",
++       main = "t distribution",
 +       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2)
@@ Line 696: / Line 671: @@
 +      pos = 4, col="red", cex=1)
 >
+</code>
+{{pasted:20260412-063722.png}}
+<code>
 > prob
-[1] 0.07001806
+[1] 0.002460977
 >
 > print(c(t.cal, df.smp, prob))
-[1]  1.91985655 19.00000000  0.07001806
+[1]  3.488086575 19.000000000  0.002460977
 > print(c(m.smp+lo2*se.z, m.smp+hi2*se.z))
-[1]  99.55202 110.38124
+[1] 102.5239 110.0970
-> cat("t =", t.cal, ", df =", round(df.smp,0), ", p-value =", prob,
+> cat(" t =", t.cal, ", df =", round(df.smp,0), ", p-value =", prob,
 + "\n", "95% confidence interval =", m.smp+lo2*se.z, m.smp+hi2*se.z)
-t = 1.919857 , df = 19 , p-value = 0.07001806
+ t = 3.488087 , df = 19 , p-value = 0.002460977
-% confidence interval = 99.55202 110.3812> t.test(smp, mu=mean(p1))
+% confidence interval = 102.5239 110.097> t.test(smp, mu=mean(p1))
 	One Sample t-test
 data:  smp
-t = 1.9199, df = 19, p-value = 0.07002
+t = 3.4881, df = 19, p-value = 0.002461
 alternative hypothesis: true mean is not equal to 100
 percent confidence interval:
-.55202 110.38124
+.5239 110.0970
 sample estimates:
 mean of x
-.9666
+.3104
 >
+> #################################
 > # t-test 2 group
-> set.seed(1996)
+> #################################
-> sz.a <- 16
+> set.seed(169)
-> sz.b <- 16
+> sz.a <- 25
+> sz.b <- 25
 > group.a <- sample(p1, sz.a)
 > group.b <- sample(p2, sz.b)
-> group.a
- [1]  80.00299 100.04410  98.51054 101.75280 113.84859  89.37281  97.89621  96.86679  89.22647 116.71786  84.21395
-[12] 109.43833 131.00954  99.24167 106.13687 110.95142
-> group.b
- [1] 102.63422 118.82094 101.30780 104.73424 107.63392 121.28520  90.89126  99.90229 116.67483  97.65544  87.06784
-[12] 102.34730  99.68162 120.60669 125.69868  98.76048
 > m.a <- mean(group.a)
 > m.b <- mean(group.b)
@@ Line 740: / Line 714: @@
 > df <- df.a+df.b
 > ss.a
-[1] 2537.948
+[1] 2225.751
 > ss.b
-[1] 1950.16
+[1] 2783.816
 > df.a
-[1] 15
+[1] 24
 > df.b
-[1] 15
+[1] 24
 >
 > pooled.v <- (ss.a+ss.b)/(df.a+df.b)
 > pooled.v
-[1] 149.6036
+[1] 104.366
 > se.s <- sqrt(pooled.v/sz.a+pooled.v/sz.b)
 > se.s
-[1] 4.324401
+[1] 2.889512
 > diff <- m.a-m.b
 > t.cal <- diff/se.s
 > t.cal
-[1] -1.01852
+[1] -3.070212
 >
 > prob <- pt(abs(t.cal), df=df, lower.tail = F)*2
 >
 > t.cal
-[1] -1.01852
+[1] -3.070212
 > df
-[1] 30
+[1] 48
 > prob
-[1] 0.3165751
+[1] 0.003515457
 >
 > t.test(group.a, group.b, var.equal = T)
@@ Line 773: / Line 747: @@
 data:  group.a and group.b
-t = -1.0185, df = 30, p-value = 0.3166
+t = -3.0702, df = 48, p-value = 0.003515
 alternative hypothesis: true difference in means is not equal to 0
 percent confidence interval:
- -13.236094   4.427118
+ -14.681167  -3.061661
 sample estimates:
 mean of x mean of y
-.5769  105.9814
+.0286  109.9000
 >
@@ Line 785: / Line 759: @@
 > hi2 <- -lo2
 > c(lo2, hi2)
-[1] -2.042272  2.042272
+[1] -2.010635  2.010635
 >
 > curve(dt(x, df=df), from = -6, to = 6,
@@ Line 800: / Line 774: @@
 +      pos=4, col='red')
 >
+</code>
+{{pasted:20260412-063739.png}}
+<code>
 > print(paste(t.cal, df, prob))
-[1] "-1.01851970325833 30 0.316575072953383"
+[1] "-3.07021182079817 48 0.00351545738746208"
 > t.test(group.a, group.b, var.equal = T)
@@ Line 807: / Line 786: @@
 data:  group.a and group.b
-t = -1.0185, df = 30, p-value = 0.3166
+t = -3.0702, df = 48, p-value = 0.003515
 alternative hypothesis: true difference in means is not equal to 0
 percent confidence interval:
- -13.236094   4.427118
+ -14.681167  -3.061661
 sample estimates:
 mean of x mean of y
-.5769  105.9814
+.0286  109.9000
 > t.cal
-[1] -1.01852
+[1] -3.070212
-> # t.cal=diff/se
-> t.cal * se.s
-[1] -4.404488
-> diff
-[1] -4.404488
-> diff+lo2*se.s
-[1] -13.23609
-> diff+hi2*se.s
-[1] 4.427118
-> (t.cal+lo2)*se.s
-[1] -13.23609
-> (t.cal+hi2)*se.s
-[1] 4.427118
 >
 > ######################
 > # 4번째 케이스 t-test
 > ######################
-> set.seed(3)
+> set.seed(37)
 > sz <- 40
 > time.a <- sample(p1,sz)
@@ Line 843: / Line 809: @@
 > diff <- m.a-m.b
 > diff
-[1] -6.116895
+[1] -8.674375
 > se.s <- sd(diff.time)/sqrt(sz)
 > t.cal <- diff/se.s
@@ Line 849: / Line 815: @@
 > prob <- pt(abs(t.cal), df=sz-1, lower.tail = F)*2
 > t.cal
-[1] -2.672942
+[1] -3.88213
 > df
 [1] 39
 > prob
-[1] 0.01092088
+[1] 0.0003888961
 > diff
-[1] -6.116895
+[1] -8.674375
 >
 > m.diff.time <- mean(diff.time)
 > se.s
-[1] 2.28845
+[1] 2.234437
 >
 > t.test(time.a, time.b, paired=T)
@@ Line 866: / Line 832: @@
 data:  time.a and time.b
-t = -2.6729, df = 39, p-value = 0.01092
+t = -3.8821, df = 39, p-value = 0.0003889
 alternative hypothesis: true mean difference is not equal to 0
 percent confidence interval:
- -10.745721  -1.488068
+ -13.193950  -4.154799
 sample estimates:
 mean difference
-      -6.116895
+      -8.674375
 >
 > m.diff.time
-[1] -6.116895
+[1] -8.674375
 > se.s
-[1] 2.28845
+[1] 2.234437
 > lo1 <- qt(0.32/2,sz-1)
 > hi1 <- -lo1
@@ Line 886: / Line 852: @@
 > hi3 <- -lo3
 >
-> curve(dt(x, df=sz-1), from = -5, to = 7,
+> curve(dt(x, df=sz-1), from = -6, to = 7,
-+       main = "t distribution curve",
++       main = "t distribution",
 +       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
 >
@@ Line 899: / Line 865: @@
 >
 > cat(t.cal, sz-1, prob)
--2.672942 39 0.01092088
+-3.88213 39 0.0003888961
->
 >
 </code>
+{{pasted:20260412-063758.png}}
+<code>
+> cat(t.cal, sz-1, prob)
+-3.88213 39 0.0003888961
+>
+</code>
 </tabbox>