--- title: "Precept 6 Problem Set" output: html_document: df_print: paged --- ```{r setup, include=FALSE} knitr::opts_chunk\$set(echo = TRUE) ``` ### Problem 1 Suppose 65% SML201 students like World Coffee better than Hoagie Haven. We selected a random sample of 20 SML201 students, and asked them which they prefer. What is the probability that more than 18 students said "World Coffee"? Write R code to compute the actual probability. Hint: * We solved a very similar problem in class on April 2 with coin tosses. Each student answer can be thought of as being like a coin toss. Decide whether you want "Heads" to represent HH or WC, and then solve the problem using `pbinom`, just like we did in class. #### Solution This is like asking about the probability of a coin's coming up heads 18 times or more out of 20 when the probability of the coin's coming up heads is 65%: ```{r} 1 - pbinom(q = 18, size = 20, prob = 0.65) ``` The answer is 0.2%. Here's another way to compute the answer: ```{r} sum(dbinom(x = c(19, 20), size = 20, prob = 0.65)) ``` ### Problem 2 In class, we saw several ways to compute the cumulative probability for the binomial distribution: we used `pbinom`; we summed up the outputs of `dbinom`; we also generated a large sample using `rbinom`, and then computed the proportion of the generated numbers that was under a certain threshold. #### Part 2(a) Write a function named `MyPbinom1`, which works just like `pbinom`. You may use `dbinom` but not `rbinom` in the function you write. ```{r} MyPbinom1 <- function(q, size, prob){ return(sum(dbinom(x = 0:q, size = size, prob = prob ))) } MyPbinom1(q = 2, size = 10, prob = 0.45) pbinom(q = 2, size = 10, prob = 0.45) ``` #### Part 2(b) Write a function named `MyPbinom2`, which works just like `pbinom`. You may use `rbinom` but not `dbinom` in the function you write. ```{r} MyPbinom2 <- function(q, size, prob){ sample <- rbinom(n = 10000000, size = size, prob = prob) return(mean(sample <= q)) } MyPbinom2(q = 2, size = 10, prob = 0.45) pbinom(q = 2, size = 10, prob = 0.45) ```