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.
hist(samp, breaks = 5)

summary(samp)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     520    1108    1368    1452    1790    2726
sd(samp)
## [1] 515.276
library('DATA606')
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normalPlot(mean = 1465, sd=450)

qqnorm(samp)
qqline(samp)

It is nearly normal distribution slighlty right skewed but not extreme. By typical i understand the average or mean of the sample. It is 1465.

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

I would expect to be similar but not identical becasue, it is randomly selected (60) out of the population and would be different for each student. However the mean would be similar, it might be a little high or low compare to the mean of the population.

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] 1321.684 1582.449

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 conditions that needs to be met are outlined below;

-Sampled observatrions must be independent. Random sampling/assignment is used and if sampling without assignment , n < 10% of the population.

-n>= 30 and population distribution should not be extremly skewed.

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(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_mean <- mean(samp)
samp_sd <- sd(samp)
samp_se <- samp_sd/sqrt(60)
samp_se
## [1] 66.52185
# for 95% confidence interval, z=1.96
samp_mean - (1.96 * samp_se)
## [1] 1321.684
samp_mean + (1.96 * samp_se)
## [1] 1582.449

Mean of the population falls within the confidence level.

  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.

The confindence interval is 95%, so i would expect 95% of these intervals to 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] 1339.885 1591.415

On your own

The figure shows this process with 50 samples, where 48 of the resulting confidence intervals contains the true mean. The proportion is 48/50 = 96% . The expectation was the confidence level to be at least 95% not 95%, so 96% is just higher confindence level.

Confidence level picked is 90% and critical value is 1.64.

# for 95% confidence interval, z=1.96
lower <- samp_mean - (1.64 * samp_se)
upper <- samp_mean + (1.64 * samp_se)
plot_ci(lower, upper, mean(population))