Foundations for statistical inference - Confidence intervals
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.
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.
- 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.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 670 1148 1535 1593 1876 3627
The typical size is 1362 sq ft, meaning specifically that out of our sample size of 60 the average sized house is 1362 sq ft. The distribution is not bell shaped and tails off.
- Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?
Another student’s sample will be different as the function grabs 60 random observations from the population. Each time the function is ran a difference group of 60 observations will be picked, however it is possible to pick the same sample just very unlikely.
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,
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] 1443.484 1741.783This 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.
- 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? Oberservations in the sample must be independent, sample must be above 30 and distribution should not be extremely skewed.
Confidence levels
- What does “95% confidence” mean? If you’re not sure, see Section 4.2.2. 95% confidence explains means 95% of observations will fall within the mean plus or minus two times the standard error.
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:
## [1] 1499.69- 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?
Our confidence does capture our population mean. Let’s say we had 100 students in a class, we would notice that about 95 of them will see the mean fall within their interval.
- 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.
We would notice a proportion of 0.95 capture the true population mean. This is because we set an interval which will capture each mean population with 95% confidence.
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:
- Obtain a random sample.
- Calculate and store the sample’s mean and standard deviation.
- Repeat steps (1) and (2) 50 times.
- Use these stored statistics to calculate many confidence intervals.
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.
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.
## [1] 1453.060 1685.307On your own
Using the following function (which was downloaded with the data set), plot all intervals. What proportion of your confidence intervals include the true population mean? Is this proportion exactly equal to the confidence level? If not, explain why.
We notice that a proportion of 0.88 confidence intervals include the true population mean. We would expect more of the intervals to include the true population mean, however if we were to increase n from the previous step then we would notice this proportion approach .95.Pick a confidence level of your choosing, provided it is not 95%. What is the appropriate critical value?
For a 98% confidence interval we would use a critical value of 2.33.
Calculate 50 confidence intervals at the confidence level you chose in the previous question. You do not need to obtain new samples, simply calculate new intervals based on the sample means and standard deviations you have already collected. Using the
plot_cifunction, plot all intervals and calculate the proportion of intervals that include the true population mean. How does this percentage compare to the confidence level selected for the intervals?
lower_vector <- samp_mean - 2.33 * samp_sd / sqrt(n)
upper_vector <- samp_mean + 2.33 * samp_sd / sqrt(n)
plot_ci(lower_vector, upper_vector, mean(population))
Using our new 98% confidence intervals we have a proportion of 1.0 which include our population mean. This is more true to the value that we expected relative to our previous example, however if we increase the sample size n for each interval, or if we increase the number of intervals we will notice this proportion approach our confidence interval.