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--- b:head_first_statistics:using_the_normal_distribution [2022/10/27 21:59] – [Exercise] hkimscil
+++ b:head_first_statistics:using_the_normal_distribution [2023/11/01 08:28] – [When to approximate the binomial distribution with the normal] hkimscil
@@ Line 288: / Line 288: @@
 ===== Exercise =====
 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
@@ Line 357: / Line 362: @@
 </code>
 pnorm
 <code>
-# Children's IQ scores are normally distributed with a
+Mean <- 100
-# mean of 100 and a standard deviation of 15. What
+Sd <- 10
-# proportion of children are expected to have an IQ between
-# 80 and 120?
-mean=100; sd=15
+# X grid for non-standard normal distribution
-lb=80; ub=120
+x <- seq(-4, 4, length = 100) * Sd + Mean
-x <- seq(-4,4,length=100)*sd + mean
+# Density function
-hx <- dnorm(x,mean,sd)
+f <- dnorm(x, Mean, Sd)
-plot(x, hx, type="n", xlab="IQ Values", ylab="",
+plot(x, f, type = "l", lwd = 2, col = "blue", ylab = "", xlab = "Weight")
-     main="Normal Distribution", axes=FALSE)
+abline(v = Mean) # Vertical line on the mean
+</code>
-i <- x >= lb & x <= ub
+{{:b:head_first_statistics:pasted:20221027-222851.png?400}}
-lines(x, hx)
-polygon(c(lb,x[i],ub), c(0,hx[i],0), col="red")
-area <- pnorm(ub, mean, sd) - pnorm(lb, mean, sd)
+<code>
-result <- paste("P(",lb,"< IQ <",ub,") =",
+# mean: mean of the Normal variable
-                signif(area, digits=3))
+# sd: standard deviation of the Normal variable
-mtext(result,3)
+# lb: lower bound of the area
-axis(1, at=seq(40, 160, 20), pos=0)
+# ub: upper bound of the area
+# acolor: color of the area
+# ...: additional arguments to be passed to lines function
+normal_area <- function(mean = 0, sd = 1, lb, ub, acolor = "lightgray", ...) {
+    x <- seq(mean - 3 * sd, mean + 3 * sd, length = 100)
+    if (missing(lb)) {
+       lb <- min(x)
+    }
+    if (missing(ub)) {
+        ub <- max(x)
+    }
+    x2 <- seq(lb, ub, length = 100)
+    plot(x, dnorm(x, mean, sd), type = "n", ylab = "")
+    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>
+<code>
+normal_area(mean = 0, sd = 1, lb = -1, ub = 2, lwd = 2)
+</code>
+{{:b:head_first_statistics:pasted:20221027-224243.png?500}}
+<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>
 </WRAP>
@@ Line 823: / Line 886: @@
 </code>
+위는 아래와 같음을 이해해야 한다
+<code>
+> sum(dbinom(c(1:5),12,1/2))
+[1] 0.3869629
+>
+</code>
 </WRAP>
@@ Line 861: / Line 930: @@
 > pnorm(-0.29)
 [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>