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.

The distribution is right skewed. The typical size within my sample is somewhere around 1387 - I am interpreting “typical” as Mean.

summary(samp)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     698    1236    1398    1467    1637    2668
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?

I don’t expect another student’s distribution to be identical to mine - 60 is not a large sample so a lot of variability is possible. I would be somewhat similar but could be skewed left instead of right and spread and mean vary.

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] 1364.615 1569.885

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 data must be sampled randomly.The sample values must be independent of each other. The sample size must be sufficiently large (n>30).

Confidence levels

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

Suppose we took many samples and built a confidence interval from each sample, then about 95% of those intervals would contain the actual mean, μ.

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. My interval was (1261, 1513).

  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.

95% of those intervals should capture our true mean since that is the definition of 95% confidence interval.

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] 1346.275 1684.559

On your own

Only 3 out of 50 intervals don’t include the true mean. So 47/50 include the true mean:

47/50*100
## [1] 94

94% include the mean. It does not exactly equal 95% because this is still an estimate - but is very close!

qnorm(0.04, lower.tail=FALSE)
## [1] 1.750686
lower_vector <- samp_mean - 1.75 * samp_sd / sqrt(n) 
upper_vector <- samp_mean + 1.75 * samp_sd / sqrt(n)

lower_vector
##  [1] 1364.397 1391.477 1321.546 1426.256 1411.923 1380.880 1358.473
##  [8] 1417.722 1389.544 1433.484 1258.095 1506.867 1327.793 1408.118
## [15] 1426.923 1444.827 1363.712 1279.244 1374.970 1468.538 1432.675
## [22] 1475.174 1343.151 1301.940 1434.685 1298.133 1440.840 1312.892
## [29] 1373.827 1374.066 1448.571 1380.621 1461.946 1467.219 1324.706
## [36] 1258.138 1337.681 1394.017 1341.180 1388.017 1276.896 1322.016
## [43] 1473.904 1426.513 1465.186 1373.170 1319.593 1376.840 1295.972
## [50] 1350.726
upper_vector
##  [1] 1666.436 1604.090 1554.088 1679.744 1634.877 1609.387 1568.493
##  [8] 1623.278 1592.889 1667.916 1460.639 1713.800 1541.541 1724.582
## [15] 1639.543 1683.507 1578.455 1543.423 1576.196 1677.029 1628.691
## [22] 1715.626 1539.416 1523.060 1628.549 1516.167 1677.960 1522.108
## [29] 1589.406 1597.401 1668.463 1601.013 1672.821 1689.114 1517.360
## [36] 1429.095 1547.419 1597.450 1532.587 1605.783 1500.771 1539.018
## [43] 1751.096 1642.854 1758.314 1602.330 1504.974 1592.394 1462.028
## [50] 1615.740
plot_ci(lower_vector, upper_vector, mean(population))

#Proportion of intervals that include true mean:

46/50*100
## [1] 92

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.