Sampling from Ames, Iowa

If you have access to data on an entire population, say the size of every house in Ames, Iowa, it’s straight forward 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.

Before we do, let’s restart R so that the latest functions added to the oilabs package get loaded. Go to the menu bar -> Session -> Restart R.

The data

In the previous lab, ``Sampling Distributions’’, we looked at the population data of houses from Ames, Iowa. Let’s start by loading that data set again.

library(mosaic)
library(oilabs)
data(ames)
head(ames)

We can rename the Gr.Liv.Area variable in the same way we did in the last lab, for consistency.

ames <- ames %>%
  rename(area = Gr.Liv.Area)

In this lab we’ll start with a simple random sample of size 60 from the population. Specifically, this is a simple random sample of size 60. 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.

population <- ames %>% 
  select(area) 
samp <- sample(population, 60)
  1. Describe the distribution of your sample. What would you say is the “typical” size within your sample? Also state precisely what you interpreted “typical” to mean.

  2. Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?

Confidence intervals

One of the most common ways to describe the typical or central value of a distribution is to use the mean. In this case we can calculate the mean of the sample using,

sample_mean <- mean(~area, data = samp)

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 sample_mean). That serves as a good point estimate but it would be useful to also communicate how uncertain we are of that estimate. This can be captured by using a confidence interval.

We can calculate a 95% confidence interval for a sample mean by adding and subtracting a particular number of standard errors. You might have that number memorized for the 95% confidence interval, but lets use qnorm() to confirm. qnorm(p) corresponds to doing a reverse look up in the normal table: for a given area/percentage p, it returns the value z such that the area under the standard normal curve to the left of z is p:

zstar <- qnorm(0.975) #why 0.975?
zstar

Now we can build the confidence interval.

s <- sd(~area, data = samp)
se <-  s / sqrt(60)

lower <- sample_mean - zstar * se
upper <- sample_mean + zstar * se

c(lower, upper)

This is an important inference that we’ve just made: 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.

  1. For the confidence interval to be valid, the sample mean must be normally distributed and have standard error \(s / \sqrt{n}\). What conditions must be met for this to be true?

Confidence levels

  1. What does “95% confidence” mean? If you’re not sure, see Section 4.2.2.

In this case we have the luxury of knowing the true population mean since we have data on the entire population. This value can be calculated using the following command:

mean(~area, data = population)
  1. Does your confidence interval capture the true average size of houses in Ames? If you are working on this lab in a classroom, does your neighbor’s interval capture this value?

  2. Each student in your class 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 recreate many samples to learn more about how sample means and confidence intervals vary from one sample to another. The do() function comes in handy here (If you are unfamiliar with do(), review the Sampling Distribution Lab).

Here is the rough outline:

We can accomplish this using the do and favstats functions in mosaic. The following lines of code takes a uniform random sample of size 60 from population, computes several summary statistics (including mean and sd), and then does this 50 times and saves the result as a data frame.

n <- 60
samp <- do(50) * favstats(~area, data = sample(population, size = n))

Have a look at the contents of samp. Lastly, we construct the confidence intervals.

samp <- samp %>%
  mutate( lower = mean - zstar * (sd / sqrt(n))) %>%
  mutate( upper = mean + zstar * (sd / sqrt(n)))

Have another look at the contents of samp. Lower bounds of these 50 confidence intervals are stored in lower, and the upper bounds are in upper. Let’s view the first interval by selecting the variables lower and upper and then sliceing the data set to take just the first set of values:

samp %>% 
  select(lower, upper) %>% 
  slice(1)

Now, we can plot how many of our confidence intervals intersect the true population mean using the plot_ci function:

with(samp, plot_ci(lower, upper, m = mean(~area, data = population)))
  1. How many of your confidence intervals contained the true mean? Is this the number you expected to see?
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.