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b:head_first_statistics:using_the_normal_distribution [2022/10/27 22:14] – [Exercise] hkimscilb:head_first_statistics:using_the_normal_distribution [2024/10/28 08:09] (current) – […then squash the width] hkimscil
Line 88: Line 88:
  
 ===== So how do we find normal probabilities? ===== ===== So how do we find normal probabilities? =====
 +평균이 0 이고 표준편차가 1일 Normal distribution 에서의 probabilities는 아래의 PDF 파일과 같이 구해 놓은 값이 있다 
 +(R을 이용하지 않는다면). [[https://ux1.eiu.edu/~aalvarado2/z_table.pdf|z table]] 링크
 +평균과 표준편차 값이 0, 1이 아닌 다른 값을 같는 분포는 0, 1 이 되도록 변환한 후에 probability를 구한다 (표준점수화). 
 +
 +
 {{:b:head_first_statistics:pasted:20191106-070200.png}} {{:b:head_first_statistics:pasted:20191106-070200.png}}
  
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 z & = & \displaystyle \frac {x - \mu}{\sigma} \\ z & = & \displaystyle \frac {x - \mu}{\sigma} \\
 & = & \frac {64-71} {4.5} \\ & = & \frac {64-71} {4.5} \\
-& = & 1.56+& = & - 1.56
 \end{eqnarray*} \end{eqnarray*}
  
-따라서, 표준점수를 1.56을 가지고 표준점수 테이블에서 1.56보다 큰 부분의 면적을 구한것을 참조하면 된다. +따라서, 표준점수를 -1.56을 가지고 표준점수 테이블에서 -1.56보다 큰 부분의 면적을 구한것을 참조하면 된다. 
  
-<code>> a <- c(1:100) +<code> 
-> scale(a) +> 1 - pnorm(-1.56) 
-              [,1] +[1] 0.9406201 
-  [1,] -1.70622042 +> pnorm(-1.56, lower.tail = FALSE) 
-  [2,] -1.67175132 +[1] 0.9406201 
-  [3,] -1.63728222 +> pnorm(64, 71, sqrt(20.25), lower.tail = FALSE) 
-  [4,] -1.60281312 +[1] 0.9400931 
-  [5,] -1.56834402 +>  
-  [6,] -1.53387492 +</code>
-  [7,] -1.49940582 +
-  [8,] -1.46493672 +
-  [9,] -1.43046762 +
- [10,] -1.39599852 +
- [11,] -1.36152943 +
- [12,] -1.32706033 +
- [13,] -1.29259123 +
- [14,] -1.25812213 +
- [15,] -1.22365303 +
- [16,] -1.18918393 +
- [17,] -1.15471483 +
- [18,] -1.12024573 +
- [19,] -1.08577663 +
- [20,] -1.05130753 +
- [21,] -1.01683843 +
- [22,] -0.98236933 +
- [23,] -0.94790023 +
- [24,] -0.91343113 +
- [25,] -0.87896203 +
- [26,] -0.84449293 +
- [27,] -0.81002384 +
- [28,] -0.77555474 +
- [29,] -0.74108564 +
- [30,] -0.70661654 +
- [31,] -0.67214744 +
- [32,] -0.63767834 +
- [33,] -0.60320924 +
- [34,] -0.56874014 +
- [35,] -0.53427104 +
- [36,] -0.49980194 +
- [37,] -0.46533284 +
- [38,] -0.43086374 +
- [39,] -0.39639464 +
- [40,] -0.36192554 +
- [41,] -0.32745644 +
- [42,] -0.29298734 +
- [43,] -0.25851825 +
- [44,] -0.22404915 +
- [45,] -0.18958005 +
- [46,] -0.15511095 +
- [47,] -0.12064185 +
- [48,] -0.08617275 +
- [49,] -0.05170365 +
- [50,] -0.01723455 +
- [51,]  0.01723455 +
- [52,]  0.05170365 +
- [53,]  0.08617275 +
- [54,]  0.12064185 +
- [55,]  0.15511095 +
- [56,]  0.18958005 +
- [57,]  0.22404915 +
- [58,]  0.25851825 +
- [59,]  0.29298734 +
- [60,]  0.32745644 +
- [61,]  0.36192554 +
- [62,]  0.39639464 +
- [63,]  0.43086374 +
- [64,]  0.46533284 +
- [65,]  0.49980194 +
- [66,]  0.53427104 +
- [67,]  0.56874014 +
- [68,]  0.60320924 +
- [69,]  0.63767834 +
- [70,]  0.67214744 +
- [71,]  0.70661654 +
- [72,]  0.74108564 +
- [73,]  0.77555474 +
- [74,]  0.81002384 +
- [75,]  0.84449293 +
- [76,]  0.87896203 +
- [77,]  0.91343113 +
- [78,]  0.94790023 +
- [79,]  0.98236933 +
- [80,]  1.01683843 +
- [81,]  1.05130753 +
- [82,]  1.08577663 +
- [83,]  1.12024573 +
- [84,]  1.15471483 +
- [85,]  1.18918393 +
- [86,]  1.22365303 +
- [87,]  1.25812213 +
- [88,]  1.29259123 +
- [89,]  1.32706033 +
- [90,]  1.36152943 +
- [91,]  1.39599852 +
- [92,]  1.43046762 +
- [93,]  1.46493672 +
- [94,]  1.49940582 +
- [95,]  1.53387492 +
- [96,]  1.56834402 +
- [97,]  1.60281312 +
- [98,]  1.63728222 +
- [99,]  1.67175132 +
-[100,]  1.70622042 +
-attr(,"scaled:center") +
-[1] 50.5 +
-attr(,"scaled:scale") +
-[1] 29.01149 +
-> aa <- scale(a) +
-> mean(aa) +
-[1] 0 +
-> sd(aa) +
-[1] 1 +
-> </code>+
  
 ==== exercise ==== ==== exercise ====
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 ===== Exercise ===== ===== Exercise =====
 Julie with 5" heels = 64 + 5 = 69 Julie with 5" heels = 64 + 5 = 69
 +Remember X ~ N(71, 20.25)
 +mean = 71
 +variance = 20.25
 +sd = 4.5
 +z = (71-69)/4.5
 z score = -0.44 z score = -0.44
  
Line 359: Line 265:
  
 <code> <code>
 +Mean <- 100
 +Sd <- 10
  
-x <- seq(-4,4, length=100) +# X grid for non-standard normal distribution 
-y <- dnorm(x) +x <- seq(-4, 4, length = 100) * Sd + Mean 
-plot(x,y, type="l")+
  
 +# Density function
 +f <- dnorm(x, Mean, Sd)
 +
 +plot(x, f, type = "l", lwd = 2, col = "blue", ylab = "", xlab = "Weight")
 +abline(v = Mean) # Vertical line on the mean
 </code> </code>
-{{:b:head_first_statistics:pasted:20221027-221410.png}} 
-<code> 
-# Children's IQ scores are normally distributed with a 
-# mean of 100 and a standard deviation of 15. What 
-# proportion of children are expected to have an IQ between 
-# 80 and 120? 
  
-mean=100; sd=15 +{{:b:head_first_statistics:pasted:20221027-222851.png?400}}
-lb=80; ub=120+
  
-x <- seq(-4,4,length=100)*sd + mean +<code> 
-hx <- dnorm(x,mean,sd)+# mean: mean of the Normal variable 
 +# sd: standard deviation of the Normal variable 
 +# lb: lower bound of the area 
 +# ub: upper bound of the area 
 +# acolor: color of the area 
 +# ...: additional arguments to be passed to lines function
  
-plot(x, hx, type="n", xlab="IQ Values", ylab="", +normal_area <- function(mean = 0, sd = 1, lb, ub, acolor = "lightgray", ...) { 
-     main="Normal Distribution", axes=FALSE)+    x <- seq(mean - 3 * sd, mean + 3 * sd, length = 100)  
 +     
 +    if (missing(lb)) { 
 +       lb <- min(x) 
 +    } 
 +    if (missing(ub)) { 
 +        ub <- max(x) 
 +    }
  
-i <- x >= lb & x <= ub +    x2 <- seq(lb, ub, length = 100)     
-lines(x, hx) +    plot(x, dnorm(x, mean, sd), type = "n", ylab = "") 
-polygon(c(lb,x[i],ub), c(0,hx[i],0), col="red")+    
 +    y <- dnorm(x2, mean, sd) 
 +    polygon(c(lb, x2, ub), c(0, y, 0), col = acolor) 
 +    lines(x, dnorm(x, mean, sd), type = "l", ...) 
 +} 
 +</code>
  
-area <- pnorm(ub, mean, sd) - pnorm(lb, mean, sd) +<code> 
-result <- paste("P(",lb,"< IQ <",ub,") =", +normal_area(mean = 0, sd = 1, lb = -1, ub = 2, lwd = 2) 
-                signif(area, digits=3)) +</code> 
-mtext(result,3) +{{:b:head_first_statistics:pasted:20221027-224243.png?500}} 
-axis(1, at=seq(40, 160, 20), pos=0)+<code> 
 +pnorm(2) 
 +pnorm(-1) 
 +pnorm(2)-pnorm(-1) 
 +ar <- round(pnorm(2)-pnorm(-1),3) 
 +</code> 
 +<code> 
 +> pnorm(2) 
 +[1] 0.9772499 
 +> pnorm(-1) 
 +[1] 0.1586553 
 +> pnorm(2)-pnorm(-1) 
 +[1] 0.8185946 
 +> ar <- round(pnorm(2)-pnorm(-1),3) 
 +>  
 +</code> 
 +<code> 
 +m.s <- 100 
 +sd.s <- 15 
 +lb <- 80 
 +ub <- 110 
 +normal_area(mean = m.s, sd = sd.s, lb = lb, ub = ub, lwd = 2) 
 +ar <- round(pnorm(ub, m.s, sd.s)-pnorm(lb, m.s, sd.s),3) 
 +text(m.s, .01, ar) 
 +</code> 
 +{{:b:head_first_statistics:pasted:20221027-225952.png?500}} 
 +<code> 
 +m.s <- 100 
 +sd.s <- 15 
 +lb <- m.s - sd.s 
 +ub <- m.s + sd.s 
 +normal_area(mean = m.s, sd = sd.s, lb = lb, ub = ub, lwd = 2) 
 +ar <- round(pnorm(ub, m.s, sd.s)-pnorm(lb, m.s, sd.s),3) 
 +text(m.s, .01, ar)
 </code> </code>
-{{:b:head_first_statistics:pasted:20221027-215953.png}} 
 </WRAP> </WRAP>
 ===== Headline ===== ===== Headline =====
Line 833: Line 787:
 </code> </code>
  
 +위는 아래와 같음을 이해해야 한다
 +<code>
 +> sum(dbinom(c(0:5),12,1/2))
 +[1] 0.387207
 +> 
 +</code>
 </WRAP> </WRAP>
  
Line 871: Line 831:
 > pnorm(-0.29) > pnorm(-0.29)
 [1] 0.3859081 [1] 0.3859081
 +
 +# the below is the same as the above
 +> n <- 12
 +> p <- 1/2
 +> q <- 1-p
 +> pnorm(5.5, n*p, sqrt(n*p*q))
 +[1] 0.386415
 +> 
 </code> </code>
  
b/head_first_statistics/using_the_normal_distribution.1666876452.txt.gz · Last modified: 2022/10/27 22:14 by hkimscil
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