Inference for categorical data
```{r global_options, include=FALSE} knitr::opts_chunk$set(eval = TRUE, results = FALSE, fig.show = "hide", message = FALSE) library(tidyverse) library(openintro) library(infer) no_helmet <- yrbss %>% filter(helmet_12m == "never") no_helmet <- no_helmet %>% mutate(text_ind = ifelse(text_while_driving_30d == "30", "yes", "no")) n <- 1000 p <- seq(from = 0, to = 1, by = 0.01) me <- 2 * sqrt(p * (1 - p)/n) dd <- data.frame(p = p, me = me) ``` ```{r sf-app, echo=FALSE, eval=TRUE, results=TRUE} inputPanel( numericInput("n", label = "Sample size:", value = 300), sliderInput("p", label = "Population proportion:", min = 0, max = 1, value = 0.1, step = 0.01), numericInput("x_min", label = "Min for x-axis:", value = 0, min = 0, max = 1), numericInput("x_max", label = "Max for x-axis:", value = 1, min = 0, max = 1) ) renderPlot({ pp <- data.frame(p_hat = rep(0, 5000)) for(i in 1:5000){ samp <- sample(c(TRUE, FALSE), input$n, replace = TRUE, prob = c(input$p, 1 - input$p)) pp$p_hat[i] <- sum(samp == TRUE) / input$n } bw <- diff(range(pp$p_hat)) / 30 ggplot(data = pp, aes(x = p_hat)) + geom_histogram(binwidth = bw) + xlim(input$x_min, input$x_max) + ggtitle(paste0("Distribution of p_hats, drawn from p = ", input$p, ", n = ", input$n)) }) ``` Answer the following question with the use of the app above. 1. Describe the sampling distribution of sample proportions at $n = 300$ and $p = 0.1$. Be sure to note the center, spread, and shape. 2. Keep $n$ constant and change $p$. How does the shape, center, and spread of the sampling distribution vary as $p$ changes. You might want to adjust min and max for the $x$-axis for a better view of the distribution. 3. Now also change $n$. How does $n$ appear to affect the distribution of $\hat{p}$?
