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.

library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
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.
hist(population)

hist(samp)

The distribution of the sample is normally distributed. Typical size means, the minimum sample size for analyzing the population distribution. Although the popuation distribution is right skewed, if we fetch decent sample size of 30, the sample will be normally distributed. And the each sample should be independent of another sample.

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

Yes, most of the student’s distribution will be identical or normally distributed. This is because of centeral theorem. It happens only if we have a decent sample (more than 30).

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)
mean(population)
## [1] 1499.69

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] 1384.644 1653.123

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?

1. Each sample should be independent. It should have equal probability of fetching. 2. If the population distribution is skewed, then the sample size should be atleast 30. So that we can use z distribution. If sample size is less than 30, then we an use the T-distribution(given that the population is normally distributed) 3. Larger the sample size, more lenient we can be.

Confidence levels

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

** We can define it as multiple different ways.

  1. 95% confidence means that we are 95% confident that the taken samples will contain the point estimate in population.
  2. The mean of the sample with two standard deviations (95% confidence interval) will contain the population mean or expected value of population. **

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?
samp <- sample(population, 60)

se <- sd(samp) / sqrt(60)
lower <- sample_mean - 1.96 * se
upper <- sample_mean + 1.96 * se
c(lower, upper)
## [1] 1397.888 1639.879
mean(population)
## [1] 1499.69

If we run the above code multiple times, we might see that the confidence interval varies for different sample sizes. But in all the cases true mean falls in that specific confidence interval. This shows that the population mean will be in the confidence interval.

  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.
samp <- select(head(arrange(ames,Gr.Liv.Area),90),Gr.Liv.Area)

samp <- sample(samp$Gr.Liv.Area, 60,replace = FALSE)

sample_mean <- mean(samp)
se <- sd(samp) / sqrt(60)
lower <- sample_mean - 2.96 * se
upper <- sample_mean + 2.96 * se
c(lower, upper)
## [1] 648.7309 732.6691
mean(population)
## [1] 1499.69

We expect that the 95% of intervals will capture the true population mean. But still there are chances of making type I error.

For example, above code is a test to find out some rare scenario. We arrange and fetch the 90 observations into a data frame(purposefully used the lower number of data). If the sample contains all the smaller values and then we figure out the confidence interval, we are not actually capturing the true population mean. Depends on confidence interval, we can say that we are 95% confident that the samples will contain 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] 1366.197 1604.436

On your own

Above code shows the most of the values are in the 95% confidence interval. Only one value is not in that interval. It may also be more than 95% confidence interval.

Above code shows the 90% confidence interval. It also calculates the critical value by calcuating the z-score value with upper and lower limit.

    plot_ci(interval$lower_vector,interval$upper_vector,m = mean(population))

Above chart shows the confidence interval of 90%. About ~10% of 60 samples does not have the true mean of the population