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t-test_summary [2026/04/12 04:25] – created hkimscilt-test_summary [2026/04/16 01:27] (current) – hkimscil
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 ################################ ################################
 +set.seed(1001)
 N.p <- 1000000 N.p <- 1000000
 m.p <- 100 m.p <- 100
Line 28: Line 29:
  
 ################################ ################################
-sz <- 10+sz <- 16
 iter <- 100000 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. # 구해볼 수 있다.
 means <- rep(NA, iter) means <- rep(NA, iter)
 for (i in 1:iter) { for (i in 1:iter) {
-  # means <- append(means, mean(sample(p1, s.size, replace = T))) 
   s1 <- sample(p1, sz, replace = T)   s1 <- sample(p1, sz, replace = T)
   means[i] <- mean(s1)   means[i] <- mean(s1)
Line 50: Line 50:
 sd.means 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)+
  
-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,  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) 
-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) lo1 <- qnorm(.32/2)
 hi1 <- -lo1 hi1 <- -lo1
Line 97: Line 85:
 c(lo3,hi3) c(lo3,hi3)
  
-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=0, col="black", lwd=2)
 abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3), abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
        col=c("red","red", "blue", "blue", "orange", "orange"),         col=c("red","red", "blue", "blue", "orange", "orange"), 
        lwd=2)        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)+mean.of.sample.a <- m.means+ 1.5*sd.means
 mean.of.sample.a mean.of.sample.a
-diff <- (mean.of.sample.a - mean(sdc))+diff <- (mean.of.sample.a - m.means)
 se.z <- sd(p1)/sqrt(sz) se.z <- sd(p1)/sqrt(sz)
 diff diff
Line 117: Line 108:
  
 curve(dnorm(x), from = -4, to = z.p2+4,  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)       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 abline(v=0, col="black", lwd=2) abline(v=0, col="black", lwd=2)
Line 129: Line 120:
  
 # 새로운 UI로 게임을 하도록 한 후 # 새로운 UI로 게임을 하도록 한 후
-# UI점수를 10명에게 구했다고 가정하고+# UI점수를 sz 명에게 구했다고 가정하고
 # 새로운 UI점수가 기존의 p1 paramter와 # 새로운 UI점수가 기존의 p1 paramter와
 # 다른지 테스트 해보라  # 다른지 테스트 해보라 
Line 137: Line 128:
 # 하면 샘플의 평균과 p1의 평균은 다르다고 판단될 것이다. # 하면 샘플의 평균과 p1의 평균은 다르다고 판단될 것이다.
 # 아래는 그럼에도 불구하고 실패하는 경우이다. # 아래는 그럼에도 불구하고 실패하는 경우이다.
-set.seed(5)+set.seed(110)
 smp <- sample(p2, sz, replace=T) smp <- sample(p2, sz, replace=T)
 m.smp <- mean(smp) m.smp <- mean(smp)
Line 149: Line 140:
  
 curve(dnorm(x), from = -4, to = z.p2+4,  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)       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 abline(v=0, col="black", lwd=2) abline(v=0, col="black", lwd=2)
Line 162: Line 154:
  
 # 같은 방법으로 했는데 성공한 경우 # 같은 방법으로 했는데 성공한 경우
-set.seed(111)+set.seed(211)
 smp <- sample(p2,sz,replace=T) smp <- sample(p2,sz,replace=T)
 m.smp <- mean(smp) m.smp <- mean(smp)
Line 173: Line 165:
 z.test(smp, mean(p1), sd(p1)) z.test(smp, mean(p1), sd(p1))
  
-z.p2 <- (mean(p2)-mean(p1))/se2+z.p2 <- (mean(p2)-mean(p1))/se.z
 z.p2 z.p2
-curve(dnorm(x), from = -4.7, to = z.p2+4,  +curve(dnorm(x), from = -5, to = z.p2+5,  
-      main = "Distribution Curve", +      main = "normalized distribution of sample means \n testing with a sample from p2 (succeeded)", 
       ylab = "Density", xlab = "z-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, 
-      main = "Distribution Curve",  
-      ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2) 
 abline(v=0, col='black', lwd=2) abline(v=0, col='black', lwd=2)
 z.cal1 z.cal1
Line 198: Line 187:
  
 # type i and type ii error # 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,  curve(dnorm(x), from = -4.7, to = z.p2+4, 
       main = "Distribution Curve",        main = "Distribution Curve", 
Line 207: Line 197:
       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
 abline(v=0, col='black', lwd=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(two, -two), col='black', lwd=2)
 abline(v=c(-z.cal1, z.cal1), col='red', lwd=2) abline(v=c(-z.cal1, z.cal1), col='red', lwd=2)
Line 234: Line 220:
 # one sample t-test # one sample t-test
 ############################ ############################
 +set.seed(99)
 sz <- 20 sz <- 20
 smp <- sample(p2, sz, replace = T) smp <- sample(p2, sz, replace = T)
Line 248: Line 235:
 m.smp+lo2*se.z m.smp+lo2*se.z
  
-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)       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
 abline(v=0, col="black", lwd=2) abline(v=0, col="black", lwd=2)
Line 264: Line 251:
 print(c(t.cal, df.smp, prob)) print(c(t.cal, df.smp, prob))
 print(c(m.smp+lo2*se.z, m.smp+hi2*se.z)) 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) "\n", "95% confidence interval =", m.smp+lo2*se.z, m.smp+hi2*se.z)
 t.test(smp, mu=mean(p1)) t.test(smp, mu=mean(p1))
  
 +#################################
 # t-test 2 group  # 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.a <- sample(p1, sz.a)
 group.b <- sample(p2, sz.b) group.b <- sample(p2, sz.b)
-group.a 
-group.b 
 m.a <- mean(group.a) m.a <- mean(group.a)
 m.b <- mean(group.b) m.b <- mean(group.b)
Line 324: Line 311:
 t.test(group.a, group.b, var.equal = T) t.test(group.a, group.b, var.equal = T)
 t.cal 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 
  
 ###################### ######################
 # 4번째 케이스 t-test # 4번째 케이스 t-test
 ###################### ######################
-set.seed(3)+set.seed(37)
 sz <- 40 sz <- 40
 time.a <- sample(p1,sz) time.a <- sample(p1,sz)
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 hi3 <- -lo3 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)       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
  
Line 380: Line 360:
  
 cat(t.cal, sz-1, prob) cat(t.cal, sz-1, prob)
 +
 </code> </code>
 <tabbox ro.hypothesis.testing> <tabbox ro.hypothesis.testing>
 <code> <code>
->  
->  
 > rm(list=ls()) > rm(list=ls())
 > rnorm2 <- function(n,mean,sd){ mean+sd*scale(rnorm(n)) } > rnorm2 <- function(n,mean,sd){ mean+sd*scale(rnorm(n)) }
Line 403: Line 382:
 >  > 
 > ################################ > ################################
 +> set.seed(1001)
 > N.p <- 1000000 > N.p <- 1000000
 > m.p <- 100 > m.p <- 100
Line 411: Line 391:
 >  > 
 > ################################ > ################################
-> sz <- 10+> sz <- 16
 > iter <- 100000 > 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. > # 구해볼 수 있다.
 > means <- rep(NA, iter) > means <- rep(NA, iter)
 > for (i in 1:iter) { > for (i in 1:iter) {
-+   # means <- append(means, mean(sample(p1, s.size, replace = T))) 
 +   s1 <- sample(p1, sz, replace = T) +   s1 <- sample(p1, sz, replace = T)
 +   means[i] <- mean(s1) +   means[i] <- mean(s1)
 + } + }
 > mean(means) > mean(means)
-[1] 99.98864+[1] 99.997
 > var(means) > var(means)
-[1] 9.964541+[1] 6.215719
 > sd(means) > sd(means)
-[1] 3.156666+[1] 2.493134
 >  > 
 > # CLT에 의하면 위이 값은 > # CLT에 의하면 위이 값은
Line 435: Line 414:
 [1] 100 [1] 100
 > ms.means > ms.means
-[1] 10+[1] 6.25
 > sd.means > 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,  > 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) > lo1 <- qnorm(.32/2)
 > hi1 <- -lo1 > hi1 <- -lo1
Line 500: Line 461:
 [1] -2.575829  2.575829 [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=0, col="black", lwd=2)
 > abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3), > abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3),
 +        col=c("red","red", "blue", "blue", "orange", "orange"),  +        col=c("red","red", "blue", "blue", "orange", "orange"), 
 +        lwd=2) +        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 > 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) > se.z <- sd(p1)/sqrt(sz)
 > diff > diff
-[1] 4.743416+[1] 3.75
 > se.z > se.z
-[1] 3.162278+[1] 2.5
 > z.score  <- diff / se.z > z.score  <- diff / se.z
 > z.score > z.score
Line 525: Line 493:
 >  > 
 > curve(dnorm(x), from = -4, to = z.p2+4,  > 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) +       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2) > abline(v=0, col="black", lwd=2)
Line 536: Line 504:
 +      pos=4, col='red') +      pos=4, col='red')
 >  > 
 +</code>
 +{{pasted:20260412-063608.png}}
 +
 +<code> 
 > # 새로운 UI로 게임을 하도록 한 후 > # 새로운 UI로 게임을 하도록 한 후
 > # UI점수를 10명에게 구했다고 가정하고 > # UI점수를 10명에게 구했다고 가정하고
Line 545: Line 517:
 > # 하면 샘플의 평균과 p1의 평균은 다르다고 판단될 것이다. > # 하면 샘플의 평균과 p1의 평균은 다르다고 판단될 것이다.
 > # 아래는 그럼에도 불구하고 실패하는 경우이다. > # 아래는 그럼에도 불구하고 실패하는 경우이다.
-> set.seed(5)+> set.seed(110)
 > smp <- sample(p2, sz, replace=T) > smp <- sample(p2, sz, replace=T)
 > m.smp <- mean(smp) > m.smp <- mean(smp)
 > m.smp > m.smp
-[1] 104.5279+[1] 104.5958
 > diff <- m.smp - mean(p1) > diff <- m.smp - mean(p1)
 > se.z <- sqrt(var(p1)/sz) > se.z <- sqrt(var(p1)/sz)
Line 555: Line 527:
 > prob1 <- pnorm(abs(z.cal1), lower.tail = F)*2 > prob1 <- pnorm(abs(z.cal1), lower.tail = F)*2
 > print(c(z.cal1, sz, prob1)) > 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.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  
- 95% CI:  93.80205 106.198> + 95% CI:  96.90102 103.099> 
 > curve(dnorm(x), from = -4, to = z.p2+4,  > 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) +       ylab = "Density", xlab = "z-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2) > abline(v=0, col="black", lwd=2)
Line 575: Line 548:
 >  > 
 >  > 
 +</code>
 +{{pasted:20260412-233253.png}}
 +
 +<code>
 > # 같은 방법으로 했는데 성공한 경우 > # 같은 방법으로 했는데 성공한 경우
-> set.seed(111)+> set.seed(211)
 > smp <- sample(p2,sz,replace=T) > smp <- sample(p2,sz,replace=T)
 > m.smp <- mean(smp) > m.smp <- mean(smp)
 > m.smp > m.smp
-[1] 110.0083+[1] 107.6795
 > diff <- m.smp - mean(p1) > diff <- m.smp - mean(p1)
 > se.z <- sqrt(var(p1)/sz) > se.z <- sqrt(var(p1)/sz)
Line 586: Line 563:
 > prob2 <- pnorm(abs(z.cal2), lower.tail = F)*2 > prob2 <- pnorm(abs(z.cal2), lower.tail = F)*2
 > print(c(z.cal2, sz, prob2)) > 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.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  
- 95% CI:  93.80205 106.198>  + 95% CI:  96.90102 103.099>  
-> z.p2 <- (mean(p2)-mean(p1))/se2+> z.p2 <- (mean(p2)-mean(p1))/se.z
 > z.p2 > 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) +       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) > abline(v=0, col='black', lwd=2)
 > z.cal1 > z.cal1
          [,1]          [,1]
-[1,] 1.431858+[1,] 2.906627
 > z.cal2 > z.cal2
-         [,1] +        [,1] 
-[1,] 3.164889+[1,] 4.85694
 > two <- qnorm(.05/2) > two <- qnorm(.05/2)
 > two > two
Line 623: Line 598:
 >  > 
 >  > 
 +</code>
 +{{pasted:20260412-233208.png}}
 +
 +<code> 
 > # type i and type ii error > # 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,  > curve(dnorm(x), from = -4.7, to = z.p2+4, 
 +       main = "Distribution Curve",  +       main = "Distribution Curve", 
 +       ylab = "Density", xlab = "z-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 = "Distribution Curve", 
 +       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2) +       ylab = "Density", xlab = "z-value", col = "blue", lwd = 2, lty=2)
 > abline(v=0, col='black', lwd=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(two, -two), col='black', lwd=2)
 > abline(v=c(-z.cal1, z.cal1), col='red', lwd=2) > abline(v=c(-z.cal1, z.cal1), col='red', lwd=2)
Line 663: Line 634:
 >  > 
 >  > 
 +</code>
 +{{pasted:20260412-233411.png}}
 +
 +<code>
 > ############################ > ############################
 > # one sample t-test > # one sample t-test
 > ############################ > ############################
 +> set.seed(99)
 > sz <- 20 > sz <- 20
 > smp <- sample(p2, sz, replace = T) > smp <- sample(p2, sz, replace = T)
Line 675: Line 651:
 > prob <- pt(t.cal, df.smp, lower.tail = F)*2 > prob <- pt(t.cal, df.smp, lower.tail = F)*2
 > se.z > se.z
-[1] 2.58698+[1] 1.809134
 > qt(.05/2, df.smp) > qt(.05/2, df.smp)
 [1] -2.093024 [1] -2.093024
Line 681: Line 657:
 > hi2 <- -lo2 > hi2 <- -lo2
 > m.smp+lo2*se.z > 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) +       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
 > abline(v=0, col="black", lwd=2) > abline(v=0, col="black", lwd=2)
Line 695: Line 671:
 +      pos = 4, col="red", cex=1) +      pos = 4, col="red", cex=1)
 >  > 
 +</code>
 +{{pasted:20260412-063722.png}}
 +
 +<code>
 > prob > prob
-[1] 0.07001806+[1] 0.002460977
 >  > 
 > print(c(t.cal, df.smp, prob)) > 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)) > 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) + "\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  
- 95% confidence interval = 99.55202 110.3812> t.test(smp, mu=mean(p1))+ 95% confidence interval = 102.5239 110.097> t.test(smp, mu=mean(p1))
  
  One Sample t-test  One Sample t-test
  
 data:  smp 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 alternative hypothesis: true mean is not equal to 100
 95 percent confidence interval: 95 percent confidence interval:
-  99.55202 110.38124+ 102.5239 110.0970
 sample estimates: sample estimates:
 mean of x  mean of x 
- 104.9666  + 106.3104 
 >  > 
 +> #################################
 > # t-test 2 group  > # 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.a <- sample(p1, sz.a)
 > group.b <- sample(p2, sz.b) > 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.a <- mean(group.a)
 > m.b <- mean(group.b) > m.b <- mean(group.b)
Line 739: Line 714:
 > df <- df.a+df.b > df <- df.a+df.b
 > ss.a > ss.a
-[1] 2537.948+[1] 2225.751
 > ss.b  > ss.b 
-[1] 1950.16+[1] 2783.816
 > df.a > df.a
-[1] 15+[1] 24
 > df.b > df.b
-[1] 15+[1] 24
 >  > 
 > pooled.v <- (ss.a+ss.b)/(df.a+df.b) > pooled.v <- (ss.a+ss.b)/(df.a+df.b)
 > pooled.v > pooled.v
-[1] 149.6036+[1] 104.366
 > se.s <- sqrt(pooled.v/sz.a+pooled.v/sz.b) > se.s <- sqrt(pooled.v/sz.a+pooled.v/sz.b)
 > se.s > se.s
-[1] 4.324401+[1] 2.889512
 > diff <- m.a-m.b > diff <- m.a-m.b
 > t.cal <- diff/se.s > t.cal <- diff/se.s
 > t.cal > t.cal
-[1] -1.01852+[1] -3.070212
 >  > 
 > prob <- pt(abs(t.cal), df=df, lower.tail = F)*2  > prob <- pt(abs(t.cal), df=df, lower.tail = F)*2 
 >  > 
 > t.cal > t.cal
-[1] -1.01852+[1] -3.070212
 > df > df
-[1] 30+[1] 48
 > prob > prob
-[1] 0.3165751+[1] 0.003515457
 >  > 
 > t.test(group.a, group.b, var.equal = T) > t.test(group.a, group.b, var.equal = T)
Line 772: Line 747:
  
 data:  group.a and group.b 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 alternative hypothesis: true difference in means is not equal to 0
 95 percent confidence interval: 95 percent confidence interval:
- -13.236094   4.427118+ -14.681167  -3.061661
 sample estimates: sample estimates:
 mean of x mean of y  mean of x mean of y 
- 101.5769  105.9814 + 101.0286  109.9000 
  
 >  > 
Line 784: Line 759:
 > hi2 <- -lo2  > hi2 <- -lo2 
 > c(lo2, hi2) > c(lo2, hi2)
-[1] -2.042272  2.042272+[1] -2.010635  2.010635
 >  > 
 > curve(dt(x, df=df), from = -6, to = 6,  > curve(dt(x, df=df), from = -6, to = 6, 
Line 799: Line 774:
 +      pos=4, col='red') +      pos=4, col='red')
 >  > 
 +
 +</code>
 +{{pasted:20260412-063739.png}}
 +
 +<code>
 > print(paste(t.cal, df, prob)) > 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) > t.test(group.a, group.b, var.equal = T)
  
Line 806: Line 786:
  
 data:  group.a and group.b 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 alternative hypothesis: true difference in means is not equal to 0
 95 percent confidence interval: 95 percent confidence interval:
- -13.236094   4.427118+ -14.681167  -3.061661
 sample estimates: sample estimates:
 mean of x mean of y  mean of x mean of y 
- 101.5769  105.9814 + 101.0286  109.9000 
  
 > t.cal > 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 > # 4번째 케이스 t-test
 > ###################### > ######################
-> set.seed(3)+> set.seed(37)
 > sz <- 40 > sz <- 40
 > time.a <- sample(p1,sz) > time.a <- sample(p1,sz)
Line 842: Line 809:
 > diff <- m.a-m.b > diff <- m.a-m.b
 > diff > diff
-[1] -6.116895+[1] -8.674375
 > se.s <- sd(diff.time)/sqrt(sz) > se.s <- sd(diff.time)/sqrt(sz)
 > t.cal <- diff/se.s > t.cal <- diff/se.s
Line 848: Line 815:
 > prob <- pt(abs(t.cal), df=sz-1, lower.tail = F)*2 > prob <- pt(abs(t.cal), df=sz-1, lower.tail = F)*2
 > t.cal > t.cal
-[1] -2.672942+[1] -3.88213
 > df > df
 [1] 39 [1] 39
 > prob > prob
-[1] 0.01092088+[1] 0.0003888961
 > diff > diff
-[1] -6.116895+[1] -8.674375
 >  > 
 > m.diff.time <- mean(diff.time) > m.diff.time <- mean(diff.time)
 > se.s > se.s
-[1] 2.28845+[1] 2.234437
 >  > 
 > t.test(time.a, time.b, paired=T) > t.test(time.a, time.b, paired=T)
Line 865: Line 832:
  
 data:  time.a and time.b 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 alternative hypothesis: true mean difference is not equal to 0
 95 percent confidence interval: 95 percent confidence interval:
- -10.745721  -1.488068+ -13.193950  -4.154799
 sample estimates: sample estimates:
 mean difference  mean difference 
-      -6.116895 +      -8.674375 
  
 >  > 
 > m.diff.time  > m.diff.time 
-[1] -6.116895+[1] -8.674375
 > se.s > se.s
-[1] 2.28845+[1] 2.234437
 > lo1 <- qt(0.32/2,sz-1) > lo1 <- qt(0.32/2,sz-1)
 > hi1 <- -lo1 > hi1 <- -lo1
Line 885: Line 852:
 > hi3 <- -lo3 > 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) +       ylab = "Density", xlab = "t-value", col = "black", lwd = 2)
 >  > 
Line 898: Line 865:
 >  > 
 > cat(t.cal, sz-1, prob) > cat(t.cal, sz-1, prob)
--2.672942 39 0.01092088 +-3.88213 39 0.0003888961
-> +
 >  > 
 </code> </code>
 +{{pasted:20260412-063758.png}}
 +
 +<code> 
 +> cat(t.cal, sz-1, prob)
 +-3.88213 39 0.0003888961
 +>  
 +</code>
 +
 </tabbox> </tabbox>
t-test_summary.1775967937.txt.gz · Last modified: by hkimscil
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