If you have access to data on an entire population, say the size of every house in Ames, Iowa, it’s straightforward to answer questions like, “How big is the typical house in Ames?” and “How much variation is there in sizes of houses?”. If you have access to only a sample of the population, as is often the case, the task becomes more complicated. What is your best guess for the typical size if you only know the sizes of several dozen houses? This sort of situation requires that you use your sample to make inference on what your population looks like.
Setting a seed: We will take some random samples and build sampling distributions in this lab, which means you should set a seed on top of your lab. If this concept is new to you, review the lab concerning probability.
In this lab we will explore the data using the dplyr package and visualize it using the ggplot2 package for data visualization. The data can be found in the companion package for the OpenIntro labs, oilabs.
Let’s load the packages.
library(dplyr)
library(ggplot2)
library(oilabs)We consider 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. Our particular focus for this lab will be all residential home sales in Ames between 2006 and 2010. This collection represents our population of interest. In this lab we would like to learn about these home sales by taking smaller samples from the full population. Let’s load the data.
data(ames)In this lab we’ll start with a simple random sample of size 10 from the population.
n <- 10
ames_samp <- sample_n(ames, n)
ames_samp # remove this line after you've looked at the data enough to understand it (no need to print!)Note that the data set has information on many housing variables, but for the first portion of the lab we’ll focus on the size of the house, represented by the variable area.
Describe the distribution of house area in your sample. What would you say is the “typical” size within your sample? Also state precisely what you interpreted “typical” to mean.
Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?
Return for a moment to the question that first motivated this lab: based on this sample, what can we infer about the population? Based only on this single sample, the best estimate of the average living area of houses sold in Ames would be the sample mean, usually denoted as \(\bar{x}\) (here we’re calling it x_bar). That serves as a good point estimate but it would be useful to also communicate how uncertain we are of that estimate. This uncertainty can be quantified using a confidence interval.
A confidence interval for a population mean is of the following form \[ \bar{x} + t^\star \frac{s}{\sqrt{n}} \]
You should by now be comfortable with calculating the mean and standard deviation of a sample in R. And we know that the sample size is 10. So the only remaining building block is finding the appropriate critical value for a given confidence level. We can use the qt function for this task, which will give the critical value associated with a given percentile under the normal distribution (the qt function is the same as iscaminvt except it doesn’t plot a curve). Remember that confidence levels and percentiles are not equivalent. For example, a 95% confidence level refers to the middle 95% of the distribution, and the critical value associated with this area will correspond to the 97.5th percentile.
We can find the critical value for a 95% confidence interal using
t_star_95 <- qt(0.975, n-1)
t_star_95which is roughly equal to the value critical value 1.96 that you’re likely familiar with by now.
Let’s finally calculate the confidence interval:
ames_samp %>%
summarise(x_bar = mean(area),
se = sd(area) / sqrt(n),
me = t_star_95 * se,
lower = x_bar - me,
upper = x_bar + me)To recap: even though we don’t know what the full population looks like, we’re 95% confident that the true average size of houses in Ames lies between the values lower and upper. There are a few conditions that must be met for this interval to be valid.
In this case we have the rare luxury of knowing the true population mean since we have data on the entire population. Let’s calculate this value so that we can determine if our confidence intervals actually capture it. We’ll store it in a data frame called parameter (short for population parameters), and name it mu.
parameter <- ames %>%
summarise(mu = mean(area))
parameterDoes your confidence interval capture the true average size of houses in Ames? Does your neighbor’s interval capture the population value?
Each student should have gotten a slightly different confidence interval. What proportion of those intervals would you expect to capture the true population mean? Why?
Using R, we’re going to collect many samples to learn more about how sample means and confidence intervals vary from one sample to another.
Here is the rough outline:
We can accomplish this using the rep_sample_n function. The following lines of code takes 50 random samples of size n from population (and remember we defined \(n = 10\) earlier), and computes the upper and lower bounds of the confidence intervals based on these samples.
ci <- ames %>%
rep_sample_n(size = n, reps = 50, replace = TRUE) %>%
summarise(x_bar = mean(area),
se = sd(area) / sqrt(n),
me = t_star_95 * se,
lower = x_bar - me,
upper = x_bar + me)Let’s view the first five intervals:
ci %>%
head(5)Next we’ll create a plot similar to the Confidence Interval applet and also to Figure 4.8 on page 175 of OpenIntro Statistics, 3rd Edition. The first step will be to create a new variable in the ci data frame that indicates whether the interval does or does not capture the true population mean. Note that capturing this value would mean the lower bound of the confidence interval is below the value and upper bound of the confidence interval is above the value. Remember that we create new variables using the mutate function.
ci <- ci %>%
mutate(capture_mu = ifelse(lower < parameter$mu & upper > parameter$mu, "yes", "no"))The ifelse function takes three arguments: first is a logical statement, second is the value we want if the logical statement yields a true result, and the third is the value we want if the logical statement yields a false result.
We now have all the information we need to create the plot. Note that the geom_errorbar() function only understands y values, and thus we have used the coord_flip() function to flip the coordinates of the entire plot back to the more familiar vertical orientation.
qplot(data = ci, x = replicate, y = x_bar, color = capture_mu) +
geom_errorbar(aes(ymin = lower, ymax = upper)) +
geom_hline(data = parameter, aes(yintercept = mu), color = "darkgray") + # draw vertical line
coord_flip()Pick a confidence level of your choosing other than 95%. What is the appropriate critical value?
Calculate 50 confidence intervals at the confidence level you chose in the previous question, and plot all intervals on one plot, and calculate the proportion of intervals that include the true population mean. How does this percentage compare to the confidence level selected for the intervals? Make sure to include your plot in your answer.
To get a better sense of the actual coverage rate, calculate 1000 intervals (don’t try to plot 1000 intervals on one graphic). How many of your intervals captured the true population value? Was it close to the confidence level you chose?
Repeat the previous question using \(z^\star\) instead of \(t^\star\). How many of your 1000 intervals captured the true population value? Was it close to the confidence level you chose? Why is the actual coverage rate with \(z^\star\) different than the coverage rate obtained using \(t^\star\)?
This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.