Foundations for statistical inference - Sampling distributions - Exercise 6
```{r global_options, include=FALSE} knitr::opts_chunk$set(eval = TRUE, results = FALSE, fig.show = "hide", message = FALSE) set.seed(1234) library(tidyverse) library(openintro) library(infer) global_monitor <- tibble( scientist_work = c(rep("Benefits", 80000), rep("Doesn't benefit", 20000)) ) samp1 <- global_monitor %>% sample_n(50) sample_props50 <- global_monitor %>% rep_sample_n(size = 50, reps = 15000, replace = TRUE) %>% count(scientist_work) %>% mutate(p_hat = n /sum(n)) %>% filter(scientist_work == "Doesn't benefit") ``` Use the app below to create sampling distributions of proportions of *Doesn't benefit* from samples of size 10, 50, and 100. Use 5,000 simulations. What does each observation in the sampling distribution represent? How does the mean, standard error, and shape of the sampling distribution change as the sample size increases? How (if at all) do these values change if you increase the number of simulations? (You do not need to include plots in your answer.) ```{r shiny, echo=FALSE, eval=TRUE, results = TRUE} shinyApp( ui <- fluidPage( # Sidebar with a slider input for number of bins sidebarLayout( sidebarPanel( selectInput("outcome", "Outcome of interest:", choices = c("Benefits", "Doesn't benefit"), selected = "Doesn't benefit"), numericInput("n_samp", "Sample size:", min = 1, max = nrow(global_monitor), value = 30), numericInput("n_rep", "Number of samples:", min = 1, max = 30000, value = 15000), hr(), sliderInput("binwidth", "Binwidth:", min = 0, max = 0.25, value = 0.01, step = 0.005) ), # Show a plot of the generated distribution mainPanel( plotOutput("sampling_plot"), textOutput("sampling_mean"), textOutput("sampling_se") ) ) ), server <- function(input, output) { # create sampling distribution sampling_dist <- reactive({ global_monitor %>% rep_sample_n(size = input$n_samp, reps = input$n_rep, replace = TRUE) %>% count(scientist_work) %>% mutate(p_hat = n /sum(n)) %>% filter(scientist_work == input$outcome) }) # plot sampling distribution output$sampling_plot <- renderPlot({ ggplot(sampling_dist(), aes(x = p_hat)) + geom_histogram(binwidth = input$binwidth) + xlim(0, 1) + labs( x = paste0("p_hat (", input$outcome, ")"), title = "Sampling distribution of p_hat", subtitle = paste0("Sample size = ", input$n_samp, " Number of samples = ", input$n_rep) ) + theme(plot.title = element_text(face = "bold", size = 16)) }) ggplot(data = sample_props50, aes(x = p_hat)) + geom_histogram(binwidth = 0.02) + labs( x = "p_hat (Doesn't benefit)", title = "Sampling distribution of p_hat", subtitle = "Sample size = 50, Number of samples = 15000" ) # mean of sampling distribution output$sampling_mean <- renderText({ paste0("Mean of sampling distribution = ", round(mean(sampling_dist()$p_hat), 2)) }) # mean of sampling distribution output$sampling_se <- renderText({ paste0("SE of sampling distribution = ", round(sd(sampling_dist()$p_hat), 2)) }) }, options = list(height = 900) ) ```
