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.

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.

load("more/ames.RData")

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 Gr.Liv.Area.

population <- ames$Gr.Liv.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. My sample appears to have a positive skew. The typical values of this sample are the mean and median: 1599 and 1546, respectively.
summary(samp)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     605    1144    1540    1552    1798    2956
hist(samp)

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

It would not be identical, but it would likely be similar. They would be similar because they are both drawn from the same population, but they would not be identical due to random sampling.

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(samp)
sample_mean
## [1] 1552.033

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 1.96 standard errors to the point estimate (See Section 4.2.3 if you are unfamiliar with this formula).

se <- sd(samp) / sqrt(60)
lower <- sample_mean - 1.96 * se
upper <- sample_mean + 1.96 * se
c(lower, upper)
## [1] 1426.957 1677.109

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?

The sample observations must be independent. The sample size is large or at least 30. The population distribution is not strongly skewed, but this condition can be relaxed with a large sample size

Confidence levels

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

We are 95% confident that the population parameter is within the interval.

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(population)
## [1] 1499.69
  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?

Yes, the CI contains the population mean

  1. 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?

95%

Why?

The correct interpretation of a CI is that if we were to take repeated samples from the same population and the 95% CI calculated for each, 95% of them would contain the population mean.

If you are working in this lab in a classroom, collect data on the intervals created by other students in the class and calculate the proportion of intervals that capture the true population mean.

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. Loops come in handy here (If you are unfamiliar with loops, review the Sampling Distribution Lab).

Here is the rough outline:

But before we do all of this, we need to first create empty vectors where we can save the means and standard deviations that will be calculated from each sample. And while we’re at it, let’s also store the desired sample size as n.

samp_mean <- rep(NA, 50)
samp_sd <- rep(NA, 50)
n <- 60

Now we’re ready for the loop where we calculate the means and standard deviations of 50 random samples.

for(i in 1:50){
  samp <- sample(population, n) # obtain a sample of size n = 60 from the population
  samp_mean[i] <- mean(samp)    # save sample mean in ith element of samp_mean
  samp_sd[i] <- sd(samp)        # save sample sd in ith element of samp_sd
}

Lastly, we construct the confidence intervals.

lower_vector <- samp_mean - 1.96 * samp_sd / sqrt(n) 
upper_vector <- samp_mean + 1.96 * samp_sd / sqrt(n)

Lower bounds of these 50 confidence intervals are stored in lower_vector, and the upper bounds are in upper_vector. Let’s view the first interval.

c(lower_vector[1], upper_vector[1])
## [1] 1430.793 1695.041

On your own

plot_ci(lower_vector, upper_vector, mean(population))

47/50
## [1] 0.94
print("No, it's not exactly equal to the confidence level. Setting aside the obvious that no whole number divided by 50 will yeild .95, the .95 proportion is an expected value, and the law of large numbers indicates that as the sample size increases, the proportion will approach the expected value.")
## [1] "No, it's not exactly equal to the confidence level. Setting aside the obvious that no whole number divided by 50 will yeild .95, the .95 proportion is an expected value, and the law of large numbers indicates that as the sample size increases, the proportion will approach the expected value."
cv <- qnorm(.95)
cv
## [1] 1.644854
lower_vector <- samp_mean - cv * samp_sd / sqrt(n) 
upper_vector <- samp_mean + cv * samp_sd / sqrt(n)

plot_ci(lower_vector, upper_vector, mean(population))

43/50
## [1] 0.86
print("It's close to the confidence level, but it is a few percentage points off.")
## [1] "It's close to the confidence level, but it is a few percentage points off."

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.