Your reproducible lab report: Before you get started, download the R Markdown template for this lab. Remember all of your code and answers go in this document:
download("http://stat.duke.edu/courses/Summer17/sta101.001-1/uploads/rmd/lab4.Rmd", destfile = "lab4.Rmd")In this lab, we investigate the ways in which the statistics from a random sample of data can serve as point estimates for population parameters. We’re interested in formulating a sampling distribution of our estimate in order to learn about the properties of the estimate, such as its distribution.
Setting a seed: We will take some random samples and build sampling distributions in this lab, which means you should set a seed on top of your lab. If this concept is new to you, review the previous lab and ask me.
We consider real estate data from the city of Ames, Iowa. The details of every real estate transaction in Ames is recorded by the City Assessor’s office. Our particular focus for this lab will be all residential home sales in Ames between 2006 and 2010. This collection represents our population of interest. In this lab we would like to learn about these home sales by taking smaller samples from the full population. Let’s load the data.
load(url("https://stat.duke.edu/~mc301/data/ames.RData"))We see that there are quite a few variables in the data set, enough to do a very in-depth analysis. For this lab, we’ll restrict our attention to just two of the variables: the above ground living area of the house in square feet (area) and the sale price (price).
We can explore the distribution of areas of homes in the population of home sales visually and with summary statistics. Let’s first create a visualization, a histogram:
qplot(data = ames, x = area, binwidth = 250, geom = "histogram")Let’s also obtain some summary statistics. Note that we can do this using the summarise function. We can calculate as many statistics as we want using this function, and just string along the results. Some of the functions below should be self explanatory (like mean, median, sd, IQR, min, and max). A new function here is the quantile function which we can use to calculate values corresponding to specific percentile cutoffs in the distribution. For example quantile(x, 0.25) will yield the cutoff value for the 25th percentile (Q1) in the distribution of x. Finding these values are useful for describing the distribution, as we can use them for descriptions like “the middle 50% of the homes have areas between such and such square feet”.
ames %>%
summarise(mu = mean(area), pop_med = median(area),
sigma = sd(area), pop_iqr = IQR(area),
pop_min = min(area), pop_max = max(area),
pop_q1 = quantile(area, 0.25), # first quartile, 25th percentile
pop_q3 = quantile(area, 0.75)) # third quartile, 75th percentileIn this lab we have access to the entire population, but this is rarely the case in real life. Gathering information on an entire population is often extremely costly or impossible. Because of this, we often take a sample of the population and use that to understand the properties of the population.
If we were interested in estimating the mean living area in Ames based on a sample, we can use the following command to survey the population.
samp1 <- ames %>%
sample_n(50)This command collects a simple random sample of size 50 from the ames dataset area, which is assigned to samp1. This is like going into the City Assessor’s database and pulling up the files on 50 random home sales. Working with these 50 files would be considerably simpler than working with all 2930 home sales.
sample_n function takes a random sample of observations (i.e. rows) from the dataset, you can still refer to the variables in the dataset with the same names. Code you used in the previous exercise will also be helpful for visualizing and summarizing the sample, however be careful to not label values mu and sigma anymore since these are sample statistics, not population parameters. You can customize the labels of any of the statistics to indicate that these come from the sample.If we’re interested in estimating the average living area in homes in Ames using the sample, our best single guess is the sample mean.
samp1 %>%
summarise(x_bar = mean(area))Depending on which 50 homes you selected, your estimate could be a bit above or a bit below the true population mean of 1499.69 square feet. In general, though, the sample mean turns out to be a pretty good estimate of the average living area, and we were able to get it by sampling less than 3% of the population.
Would you expect the mean of your sample to match the mean of another student’s sample? Why, or why not? If the answer is no, would you expect the means to just be somewhat different or very different? Ask a neighboring student to confirm your answer.
Take a second sample, also of size 50, and call it samp2. How does the mean of samp2 compare with the mean of samp1? Suppose we took two more samples, one of size 100 and one of size 1000. Which would you think would provide a more accurate estimate of the population mean?
Not surprisingly, every time we take another random sample, we get a different sample mean. It’s useful to get a sense of just how much variability we should expect when estimating the population mean this way. The distribution of sample means, called the sampling distribution, can help us understand this variability. In this lab, because we have access to the population, we can build up the sampling distribution for the sample mean by repeating the above steps many times. Here we will generate 2000 samples and compute the sample mean of each. Note that we are sampling with replacement, replace = TRUE since sampling distributions are constructed with sampling with replacement.
sample_means50 <- do(2000) * ames %>%
sample_n(50, replace = TRUE) %>%
summarise(x_bar = mean(area))
qplot(data = sample_means50, x = x_bar)Here we use R to take 2000 samples of size 50 from the population, calculate the mean of each sample, and store each result in a vector called sample_means50. On the next page, we’ll review how this set of code works.
sample_means50? Describe the sampling distribution, and be sure to specifically note its center. Make sure to include a plot of the distribution in your answer.do functionLet’s take a break from the statistics for a moment to let that last block of code sink in. The idea behind the do function is repetition: it allows you to execute a line of code as many times as you want and put the results in a data frame. In the case above, we wanted to repeatedly take a random sample of size 50 from area and then save the mean of that sample into the sample_means50 vector.
Without the do function, this would be painful. First, we’d have to create an empty vector filled with 0s to hold the 2000 sample means. Then, we’d have to compute each of the 2000 sample means one line at a time, putting them individually into the slots of the sample_means50 vector:
sample_means50 <- rep(NA, 2000)
sample_means50[1] <- ames %>%
sample_n(50, replace = TRUE) %>%
summarise(x_bar = mean(area))
sample_means50[2] <- ames %>%
sample_n(50, replace = TRUE) %>%
summarise(x_bar = mean(area))
sample_means50[3] <- ames %>%
sample_n(50, replace = TRUE) %>%
summarise(x_bar = mean(area))
sample_means50[4] <- ames %>%
sample_n(50, replace = TRUE) %>%
summarise(x_bar = mean(area))
# ...and so on, 2000 timesWith the do function, these thousands of lines of code are compressed into one line:
sample_means50 <- do(2000) * ames %>%
sample_n(50, replace = TRUE) %>%
summarise(x_bar = mean(area))Note that for each of the 2.000 times we computed a mean, we did so from a different sample!
sample_n and do function do, try modifying the code to create a sampling distribution of 25 sample means from samples of size 10, and put them in a data frame named sample_means_small. Print the output. How many observations are there in this object called sample_means_small? What does each observation represent?Mechanics aside, let’s return to the reason we used the do function: to compute a sampling distribution, specifically, this one.
qplot(data = sample_means50, x = x_bar, geom = "histogram")The sampling distribution that we computed tells us much about estimating the average living area in homes in Ames. Because the sample mean is an unbiased estimator, the sampling distribution is centered at the true average living area of the population, and the spread of the distribution indicates how much variability is induced by sampling only 50 home sales.
In the remainder of this section we will work on getting a sense of the effect that sample size has on our sampling distribution.
areas from samples of size 10, 50, and 100. Use 5,000 simulations. What does each observation in the sampling distribution represent? How does the mean, standard error, and shape of the sampling distribution change as the sample size increases? How (if at all) do these values change if you increase the number of simulations? (You do not need to include plots in your answer.)So far, we have only focused on estimating the mean living area in homes in Ames. Now you’ll try to estimate the mean home price.
Take a sample of size 15 from the population and calculate the mean price of the homes in this sample. Using this sample, what is your best point estimate of the population mean of prices of homes?
Since you have access to the population, simulate the sampling distribution for \(\bar{x}_{price}\) by taking 2000 samples from the population of size 15 and computing 2000 sample means. Store these means in a vector called sample_means15. Plot the data, then describe the shape of this sampling distribution. Based on this sampling distribution, what would you guess the mean home price of the population to be? Finally, calculate and report the population mean.
Change your sample size from 15 to 150, then compute the sampling distribution using the same method as above, and store these means in a new vector called sample_means150. Describe the shape of this sampling distribution, and compare it to the sampling distribution for a sample size of 15. Based on this sampling distribution, what would you guess to be the mean sale price of homes in Ames?
Of the sampling distributions from 2 and 3, which has a smaller spread? If we’re concerned with making estimates that are more often close to the true value, would we prefer a sampling distribution with a large or small spread?
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.