Foundations for statistical inference - Confidence intervals
```{r global_options, include=FALSE} knitr::opts_chunk$set(eval = TRUE, results = FALSE, fig.show = "hide", message = FALSE) library(tidyverse) library(openintro) library(infer) us_adults <- tibble( climate_change_affects = c(rep("Yes", 62000), rep("No", 38000)) ) n <- 60 samp <- us_adults %>% sample_n(size = n) set.seed(123) us_adults %>% sample_n(size = n) %>% specify(response = climate_change_affects, success = "Yes") %>% calculate(stat = "prop") %>% #assume(distribution = "z")%>% get_ci(level = 0.95) ``` 1. Given a sample size of 60, 1000 bootstrap samples for each interval, and 50 confidence intervals constructed (the default values for the above app), what proportion of your confidence intervals include the true population proportion? Is this proportion exactly equal to the confidence level? If not, explain why. Make sure to include your plot in your answer. ```{r shiny, echo=FALSE, eval=TRUE, results = TRUE} store_ci <- function(i, n, reps, conf_level, success) { us_adults %>% sample_n(size = n) %>% specify(response = climate_change_affects, success = success) %>% generate(reps, type = "bootstrap") %>% calculate(stat = "prop") %>% get_ci(level = conf_level) %>% rename( x_lower = names(.)[1], x_upper = names(.)[2] ) } shinyApp( ui <- fluidPage( h4("Confidence intervals for the proportion of US adults who think climate change"), h4(selectInput("success", "", choices = c( "is affecting their local community" = "Yes", "is not affecting their local community" = "No" ), selected = "Yes", width = "50%" )), # Sidebar with a slider input for number of bins sidebarLayout( sidebarPanel( numericInput("n_samp", "Sample size for a single sample from the population:", min = 1, max = 1000, value = 60 ), hr(), numericInput("n_rep", "Number of resamples for each bootstrap confidence interval:", min = 1, max = 15000, value = 1000 ), numericInput("conf_level", "Confidence level", min = 0.01, max = 0.99, value = 0.95, step = 0.05 ), hr(), radioButtons("n_ci", "Number of confidence intervals:", choices = c(10, 25, 50, 100), selected = 50, inline = TRUE ), actionButton("go", "Go") ), # Show a plot of the generated distribution mainPanel( plotOutput("ci_plot") ) ) ), server <- function(input, output) { # set true p p <- reactive(ifelse(input$success == "Yes", 0.62, 0.38)) # create df_ci when go button is pushed df_ci <- eventReactive(input$go, { map_dfr(1:input$n_ci, store_ci, n = input$n_samp, reps = input$n_rep, conf_level = input$conf_level, success = input$success ) %>% mutate( y_lower = 1:input$n_ci, y_upper = 1:input$n_ci, capture_p = ifelse(x_lower < p() & x_upper > p(), "Yes", "No") ) }) # plot df_ci output$ci_plot <- renderPlot({ ggplot(df_ci()) + geom_segment(aes(x = x_lower, y = y_lower, xend = x_upper, yend = y_upper, color = capture_p)) + geom_point(aes(x = x_lower, y = y_lower, color = capture_p)) + geom_point(aes(x = x_upper, y = y_upper, color = capture_p)) + geom_vline(xintercept = p(), color = "darkgray") + labs( y = "", x = "Bounds of the confidence interval", color = "Does the interval capture the true population proportion?" ) + theme(legend.position = "bottom") }) }, options = list(height = 700) ) ```
