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.
summary(samp)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     708    1118    1512    1551    1744    4676
hist(samp, breaks = 25, xlim = range(samp))

The population is multimodal and skewed to the right. Most of the house are below 2000 sq. ft.. The typical size are between 1200 to 1400 and 1400 to 1800 sq. ft. since the amount of houses with the size in these two range are higher than the amount of other houses.
  1. Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?
length(population)
## [1] 2930
Because the sample size 60 is relative small comparing to the size of the population. 2930, I think the distribution from another student will not be identical to mine though there is occations that distributions from two sampling are similar.

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)

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] 1392.340 1709.793

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?

    In order to make sure the sample mean is normally distributed the following condeions must be met:
    1. the observations are independent and less than 10% of the population;
    2. the sample size is at least 30;
    3. the data is not strongly skewed.

Confidence levels

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

    “95% confidence” means we are 95% confident that the confident interval will contain the true population mean.

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?
# Whether population mean is in the CI?
if (mean(population) > lower & mean(population) < upper){
  print("TRUE")
} else{
  print("FALSE")
}
## [1] "TRUE"

  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? Why? 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.

    There will be 95% of the confident intervals got from the students capture the population mean. Because this is the definition of confident interval. 95% confident interval will be +/-1.96 times of standard error away from the estimation 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] 1340.343 1583.557

On your own

plot_ci(lower_vector, upper_vector, mean(population))

# What proportion of your confidence intervals include the true population mean?
CI <- list()
for (i in 1:50){
  if (mean(population) > lower_vector[i] & mean(population) < upper_vector[i]){
  CI[i] <- TRUE
  } else{
    CI[i] <- FALSE
  }
}
library(tidyr)
## Warning: package 'tidyr' was built under R version 3.3.3
## 
## Attaching package: 'tidyr'
## The following object is masked _by_ '.GlobalEnv':
## 
##     population
CI <- as.data.frame(CI)
CI <- gather(CI,"No", "TorF",1:50) # conver to long table to put results into one variable
fre <- by(CI$TorF, CI$TorF,length) # calculate the frequncy of T and F repecitively
fre
## CI$TorF: FALSE
## [1] 2
## -------------------------------------------------------- 
## CI$TorF: TRUE
## [1] 48
if(fre[1] == 50){
  paste("The proportion of the confidence intervals include the true population mean is ",pro*100,"%.")
}else{
  pro <- fre[2]/(fre[1] + fre[2])
paste("The proportion of the confidence intervals include the true population mean is",pro*100,"%.")
}
## [1] "The proportion of the confidence intervals include the true population mean is 96 %."
   This proportion is not equal to the confidence level. 95% confidence interval means that at least 95% of the sample control intervals will include the true mean. 
samp_mean <- rep(NA, 50)
samp_sd <- rep(NA, 50)
n <- 60

for(i in 1:50){
  samp <- sample(population, n)  
  samp_mean[i] <- mean(samp)     
  samp_sd[i] <- sd(samp)         
}

lower_vector <- samp_mean - 2.58 * samp_sd / sqrt(n) # 99% confident level
upper_vector <- samp_mean + 2.58 * samp_sd / sqrt(n)

plot_ci(lower_vector, upper_vector, mean(population))

# What proportion of your confidence intervals include the true population mean?
CI <- list()
for (i in 1:50){
  if (mean(population) > lower_vector[i] & mean(population) < upper_vector[i]){
  CI[i] <- TRUE
  } else{
    CI[i] <- FALSE
  }
}
library(tidyr)
CI <- as.data.frame(CI)
CI <- gather(CI,"No", "TorF",1:50) # conver to long table to put results into one variable
fre <- by(CI$TorF, CI$TorF,length) # calculate the frequncy of T and F repecitively
fre
## CI$TorF: FALSE
## [1] 1
## -------------------------------------------------------- 
## CI$TorF: TRUE
## [1] 49
if(fre[1] == 50){
  paste("The proportion of the confidence intervals include the true population mean is ",pro*100,"%.")
}else{
  pro <- fre[2]/(fre[1] + fre[2])
paste("The proportion of the confidence intervals include the true population mean is",pro*100,"%.")
}
## [1] "The proportion of the confidence intervals include the true population mean is 98 %."

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.