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--- b:head_first_statistics:using_the_normal_distribution [2022/10/27 21:43] – [Exercise] hkimscil
+++ b:head_first_statistics:using_the_normal_distribution [2023/11/01 08:29] (current) – [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 314: / Line 319: @@
 > set.seed(101)
 > rnorm(5, mean=100, sd=10)
-[1] 105.26448  92.05156 114.27756  85.33180  97.63317
+[1]  96.73964 105.52462  93.25056 102.14359 103.10769
+>
+</code>
+<code>
+> set.seed(101)
+> s1 <- rnorm(100, mean=100, sd=10)
+> s1
+  [1]  96.73964 105.52462  93.25056 102.14359 103.10769 111.73966 106.18790
+  [8]  98.87266 109.17028  97.76741 105.26448  92.05156 114.27756  85.33180
+ [15]  97.63317  98.06662  91.50245 100.58465  91.82330  79.49692  98.36244
+ [22] 107.08522  97.32019  85.36078 107.44436  85.89610 104.67068  98.80680
+ [29] 104.67239 104.98136 108.94937 102.79152 110.07866  79.26894 111.89853
+ [36]  92.75626 101.67984 109.20335  83.28395 104.48469 104.82459 107.58214
+ [43]  76.80673  95.40495  88.94616 104.02928 105.68935  92.93917  97.09909
+ [50]  85.16122  88.49745  97.25529 105.77901  86.03097 107.49058  89.48813
+ [57] 101.65381 111.29809 111.73722  95.72137  97.40198  85.88827  93.58642
+ [64] 101.12458 104.22604 103.86835  93.12202 101.48902  99.42350  99.25177
+ [71] 115.09897 116.19937 111.53158  99.22396  81.81065  89.62555 103.02492
+ [78]  87.22054 101.38339  99.49016 118.52148 111.11675  94.88625  94.56119
+ [85]  82.71073 104.70750 100.05387 113.48046 107.24097 115.52549 113.25470
+ [92]  99.65735  96.38987  92.79835 102.82015  92.09474  95.55095 113.64993
+ [99] 104.97454  91.85604
+>
+> mean(s1)
+[1] 99.62809
+> sd(s1)
+[1] 9.34071
 >
 </code>
 pnorm
 qnorm
+dnorm
+<code>
+> set.seed(101)
+> dnorm(0)
+[1] 0.3989423
+> dnorm(0, mean=0, sd=1)
+[1] 0.3989423
+> dnorm(0, mean=0, sd=5)
+[1] 0.07978846
+>
+</code>
+pnorm
+<code>
+Mean <- 100
+Sd <- 10
+# X grid for non-standard normal distribution
+x <- seq(-4, 4, length = 100) * Sd + Mean
+# 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>
+{{:b:head_first_statistics:pasted:20221027-222851.png?400}}
+<code>
+# 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
+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>
 ===== Headline =====
@@ Line 760: / Line 886: @@
 </code>
+위는 아래와 같음을 이해해야 한다
+<code>
+> sum(dbinom(c(0:5),12,1/2))
+[1] 0.387207
+>
+</code>
 </WRAP>
@@ Line 798: / 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>