In jbryer/DATA606: DATA606 Statistics and Probability for Data Analytics

knitr::opts_chunk$set(eval = FALSE)
library(tidyverse)
library(openintro)
library(infer)

In this lab, you will investigate the ways in which the statistics from a random sample of data can serve as point estimates for population parameters. We're interested in formulating a sampling distribution of our estimate in order to learn about the properties of the estimate, such as its distribution.

**Setting a seed:** We will take some random samples and build sampling distributions in this lab, which means you should set a seed at the start of your lab. If this concept is new to you, review the lab concerning probability.

Getting Started

Load packages

In this lab, we will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro. Lastly, we will use the infer package for resampling.

Let's load the packages.

library(tidyverse)
library(openintro)
library(infer)

Creating a reproducible lab report

To create your new lab report, start by opening a new R Markdown document... From Template... then select Lab Report from the openintro package.

The data

You will be analyzing real estate data from the city of Ames, Iowa. The details of every real estate transaction in Ames is recorded by the City Assessor's office. Your particular focus for this lab will be all residential home sales in Ames between 2006 and 2010. This collection represents your population of interest. In this lab, you will learn about these home sales by taking smaller samples from the full population. Let's load the data.

data(ames)

You can see that there are quite a few variables in the data set, enough to do a very in-depth analysis. For this lab, you will focus on just two of the variables: the above ground living area of the house in square feet (area) and the sale price (price).

You can explore the distribution of areas of homes in the population of home sales visually and with summary statistics. Let's first create a visualization, a histogram:

ggplot(data = ames, aes(x = area)) +
  geom_histogram(binwidth = 250)

Let's also obtain some summary statistics. Note that you can do this using the summarise function. You can calculate as many statistics as you want using this function, and just combine the results. Some of the functions below should be self explanatory (like mean, median, sd, IQR, min, and max). A new function here is the quantile function, which you can use to calculate values corresponding to specific percentile cutoffs in the distribution. For example quantile(x, 0.25) will yield the cutoff value for the 25th percentile (Q1) in the distribution of x. Finding these values is useful for describing the distribution, as you can use them for descriptions like "the middle 50% of the homes have areas between such and such square feet".

ames %>%
  summarise(mu = mean(area), pop_med = median(area), 
            sigma = sd(area), pop_iqr = IQR(area),
            pop_min = min(area), pop_max = max(area),
            pop_q1 = quantile(area, 0.25),  # first quartile, 25th percentile
            pop_q3 = quantile(area, 0.75))  # third quartile, 75th percentile

1. Describe this population distribution using a visualization and the summary statistics listed above. You don't have to use all of the summary statistics in your description, and you will need to decide which ones are relevant based on the shape of the distribution. Make sure to include the plot and the summary statistics output in your report along with your narrative.

The unknown sampling distribution

In this lab, you have access to the entire population, but this is rarely the case in real life. Gathering information on an entire population is often extremely costly or impossible. Because of this, we often take a sample of the population and use that to understand the properties of the population.

If you were interested in estimating the mean living area in Ames based on a sample, you can use the sample_n command to survey the population.

samp1 <- ames %>%
  sample_n(50)

This command collects a simple random sample of size 50 from the ames dataset, and assigns the result to samp1. This is similar to going into the City Assessor's database and pulling up the files on 50 random home sales. Working with these 50 files is considerably simpler than working with all 2930 home sales.

1. Describe the distribution of area in this sample. How does it compare to the distribution of the population? Hint: Although the sample_n function takes a random sample of observations (i.e. rows) from the dataset, you can still refer to the variables in the dataset with the same names. Code you used in the previous exercise will also be helpful for visualizing and summarizing the sample, however be careful to not label values mu and sigma anymore since these are sample statistics, not population parameters. You can customize the labels of any of the statistics to indicate that these come from the sample.

If you're interested in estimating the average living area in homes in Ames using the sample, your best single guess is the sample mean.

samp1 %>%
  summarise(x_bar = mean(area))

Depending on which 50 homes you selected, your estimate could be a bit above or a bit below the true population mean of r round(mean(ames$area),2) square feet. In general, though, the sample mean turns out to be a pretty good estimate of the average living area, and you were able to get it by sampling less than 3\% of the population.

1. Would you expect the mean of your sample to match the mean of another team's sample? Why, or why not? If the answer is no, would you expect the means to just be somewhat different or very different? Ask a neighboring team to confirm your answer.

2. Take a second sample, also of size 50, and call it samp2. How does the mean of samp2 compare with the mean of samp1? Suppose we took two more samples, one of size 100 and one of size 1000. Which would you think would provide a more accurate estimate of the population mean?

Not surprisingly, every time you take another random sample, you get a different sample mean. It's useful to get a sense of just how much variability you should expect when estimating the population mean this way. The distribution of sample means, called the sampling distribution (of the mean), can help you understand this variability. In this lab, because you have access to the population, you can build up the sampling distribution for the sample mean by repeating the above steps many times. Here, you will generate 15,000 samples and compute the sample mean of each. Note that we specify that replace = TRUE since sampling distributions are constructed by sampling with replacement.

sample_means50 <- ames %>%
                    rep_sample_n(size = 50, reps = 15000, replace = TRUE) %>%
                    summarise(x_bar = mean(area))

ggplot(data = sample_means50, aes(x = x_bar)) +
  geom_histogram(binwidth = 20)

Here, we use R to take 15,000 different samples of size 50 from the population, calculate the mean of each sample, and store each result in a vector called sample_means50. Next, you will review how this set of code works.

1. How many elements are there in sample_means50? Describe the sampling distribution, and be sure to specifically note its center. Make sure to include a plot of the distribution in your answer.

Interlude: Sampling distributions

The idea behind the rep_sample_n function is repetition. Earlier, you took a single sample of size n (50) from the population of all houses in Ames. With this new function, you can repeat this sampling procedure rep times in order to build a distribution of a series of sample statistics, which is called the sampling distribution.

Note that in practice one rarely gets to build true sampling distributions, because one rarely has access to data from the entire population.

Without the rep_sample_n function, this would be painful. We would have to manually run the following code 15,000 times

ames %>%
  sample_n(size = 50) %>%
  summarise(x_bar = mean(area))

as well as store the resulting sample means each time in a separate vector.

Note that for each of the 15,000 times we computed a mean, we did so from a different sample!

1. To make sure you understand how sampling distributions are built, and exactly what the rep_sample_n function does, try modifying the code to create a sampling distribution of 25 sample means from samples of size 10, and put them in a data frame named sample_means_small. Print the output. How many observations are there in this object called sample_means_small? What does each observation represent?

Sample size and the sampling distribution

Mechanics aside, let's return to the reason we used the rep_sample_n function: to compute a sampling distribution, specifically, the sampling distribution of the mean home area for samples of 50 houses.

ggplot(data = sample_means50, aes(x = x_bar)) +
  geom_histogram(binwidth = 20)

The sampling distribution that you computed tells you much about estimating the average living area in homes in Ames. Because the sample mean is an unbiased estimator, the sampling distribution is centered at the true average living area of the population, and the spread of the distribution indicates how much variability is incurred by sampling only 50 home sales.

In the remainder of this section, you will work on getting a sense of the effect that sample size has on your sampling distribution.

1. Use the app below to create sampling distributions of means of areas 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.)

shinyApp(
  ui <- fluidPage(

    # Sidebar with a slider input for number of bins 
    sidebarLayout(
      sidebarPanel(

        selectInput("selected_var",
                    "Variable:",
                    choices = list("area", "price"),
                    selected = "area"),         

        numericInput("n_samp",
                     "Sample size:",
                     min = 1,
                     max = nrow(ames),
                     value = 30),

        numericInput("n_sim",
                     "Number of samples:",
                     min = 1,
                     max = 30000,
                     value = 15000) 

      ),

      # Show a plot of the generated distribution
      mainPanel(
        plotOutput("sampling_plot"),
        verbatimTextOutput("sampling_mean"),
        verbatimTextOutput("sampling_se")
      )
    )
  ),

  # Define server logic required to draw a histogram
  server <- function(input, output) {

    # create sampling distribution
    sampling_dist <- reactive({
      ames[[input$selected_var]] %>%
        sample(size = input$n_samp * input$n_sim, replace = TRUE) %>%
        matrix(ncol = input$n_samp) %>%
        rowMeans() %>%
        data.frame(x_bar = .)
    })

    # plot sampling distribution
    output$sampling_plot <- renderPlot({
      x_min <- quantile(ames[[input$selected_var]], 0.1)
      x_max <- quantile(ames[[input$selected_var]], 0.9)

      ggplot(sampling_dist(), aes(x = x_bar)) +
        geom_histogram() +
        xlim(x_min, x_max) +
        ylim(0, input$n_sim * 0.35) +
        ggtitle(paste0("Sampling distribution of mean ", 
                       input$selected_var, " (n = ", input$n_samp, ")")) +
        xlab(paste("mean", input$selected_var)) +
        theme(plot.title = element_text(face = "bold", size = 16))
    })

    # mean of sampling distribution
    output$sampling_mean <- renderText({
      paste0("mean of sampling distribution = ", round(mean(sampling_dist()$x_bar), 2))
    })

    # mean of sampling distribution
    output$sampling_se <- renderText({
      paste0("SE of sampling distribution = ", round(sd(sampling_dist()$x_bar), 2))
    })
  },

  options = list(height = 500) 
)

---

More Practice

So far, you have only focused on estimating the mean living area in homes in Ames. Now, you'll try to estimate the mean home price.

Note that while you might be able to answer some of these questions using the app, you are expected to write the required code and produce the necessary plots and summary statistics. You are welcome to use the app for exploration.

1. Take a sample of size 15 from the population and calculate the mean price of the homes in this sample. Using this sample, what is your best point estimate of the population mean of prices of homes?

2. Since you have access to the population, simulate the sampling distribution of $\overline{price}$ for samples of size 15 by taking 2000 samples from the population of size 15 and computing 2000 sample means. Store these means in a vector called sample_means15. Plot the data, then describe the shape of this sampling distribution. Based on this sampling distribution, what would you guess the mean home price of the population to be? Finally, calculate and report the population mean.

3. Change your sample size from 15 to 150, then compute the sampling distribution using the same method as above, and store these means in a new vector called sample_means150. Describe the shape of this sampling distribution and compare it to the sampling distribution for a sample size of 15. Based on this sampling distribution, what would you guess to be the mean sale price of homes in Ames?

4. Of the sampling distributions from 2 and 3, which has a smaller spread? If you're concerned with making estimates that are more often close to the true value, would you prefer a sampling distribution with a large or small spread?

This is a product of OpenIntro that is released under a [Creative Commons Attribution-ShareAlike 3.0 Unported](http://creativecommons.org/licenses/by-sa/3.0). This lab was written for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.

jbryer/DATA606 documentation built on Feb. 17, 2024, 4:13 a.m.