Solved problem 9.

9/script.R

@@ -0,0 +1,76 @@
+#!/bin/env Rscript
+
+# Libraries
+library("pracma")
+library("ggplot2")
+
+# Set the seed for the RNG
+set.seed(1549)
+
+# All the `n`s we'll be iterating over
+all_n <- list(30, 50, 100, 200, 300, 500, 1000)
+all_n <- sapply(all_n, function(n) n)
+print(all_n)
+
+# Calculate the samples
+p <- 0.8
+samples_len <- 2000
+samples <- lapply(all_n, function(n) {
+    lapply(1:samples_len, function(sample_idx) {
+        rbinom(n, size = 1, prob = p)
+    })
+})
+
+# Calculate the interval lengths for each method
+confidence <- 0.98
+z <- qnorm((1.0 + confidence) / 2.0)
+method1_length <- function(mean, n) {
+    a <- z^2 / n
+    b <- a * (a + 4 * mean - 4 * mean^2)
+
+    p_min <- (2 * mean + a - sqrt(b)) / (2 * (a + 1))
+    p_max <- (2 * mean + a + sqrt(b)) / (2 * (a + 1))
+
+    p_max - p_min
+}
+
+method2_length <- function(mean, n) {
+    a <- z^2 * (mean * (1 - mean)) / n
+
+    p_min <- -sqrt(a) + mean
+    p_max <- sqrt(a) + mean
+
+    p_max - p_min
+}
+
+lengths1 <- mapply(function(samples_n, n) {
+    mean(sapply(samples_n, function(sample) {
+        method1_length(mean(sample), n)
+    }))
+}, samples, all_n)
+print(lengths1)
+
+lengths2 <- mapply(function(samples_n, n) {
+    mean(sapply(samples_n, function(sample) {
+        method2_length(mean(sample), n)
+    }))
+}, samples, all_n)
+print(lengths2)
+
+
+# Finally plot and save it
+df <- data.frame(
+    all_n = all_n,
+    lengths1 = lengths1,
+    lengths2 = lengths2
+)
+
+plot <- ggplot(df, aes(x = all_n)) +
+    geom_line(aes(y = lengths1, color = "Method 1")) +
+    geom_line(aes(y = lengths2, color = "Method 2")) +
+    xlab("N") +
+    ylab("") +
+    scale_color_manual(values = c("red", "blue")) +
+    theme(legend.title = element_blank())
+
+ggsave(plot = plot, "output.png")

README.md

@@ -16,6 +16,10 @@ https://docs.google.com/document/d/1DnvOxo3GlUY7sBHUCcyMEbMlQE2AYFBWAOSrNZyb4d4
 
 https://docs.google.com/document/d/1x_AsfHc4JNxV8P6oOfUoZCu1yLQ9TQ82zRSoT7pDRrM
 
+## 9
+
+https://docs.google.com/document/d/1TmL1dgG8DoG6-_J2rQPIU_SOcNpKvbDBL0KVmuLRe1g
+
 # Util
 
 ## 6