5 Confidence Intervals - Foundations for Statistical Inference | OpenIntro Statistics: Labs for R

Bray, Andrew; Çetinkaya-Rundel, Mine; Kuyper, Arend

5 Confidence Intervals - Foundations for Statistical Inference

This lab is structured to guide you through an organized process such that you could easily organize your code with comments — meaning your R script — into a lab report. I would suggest getting into the habit of writing an organized and commented R script that completes the tasks and answers the questions provided in the lab – including in the Own Your Own section.

5.1 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.

5.2 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.

download.file("http://www.openintro.org/stat/data/ames.RData", destfile = "ames.RData")
load("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)

Exercise 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.

Exercise 2: Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?

5.3 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). Note that the 1.96 is the result of rounding and we could ues R to find a more precise value which is provided in the code below.

qnorm(0.975) # or
qnorm(0.025) # which is the negative version

Note that if we take 0.975 - 0.025 we get 0.95. Each tail is set to have area 0.025. Usually two decimal places of accuracy is sufficient when determining the appropriate \(z^*\) value for a given confidence level. Therefore we will continue using 1.96, but keep the function above in mind when you desire to use a different confidence level or you need more precision.

se <- sd(samp) / sqrt(60)
lower <- sample_mean - 1.96 * se
upper <- sample_mean + 1.96 * se
c(lower, upper)

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.

Exercise 3: 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?

5.4 Confidence levels

Exercise 4: What does “95% confidence” mean?

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)

Exercise 5: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?

Exercise 6: 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.

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 Foundations for Statistical Inference (Lab 04)).

Here is the rough outline:

Obtain a random sample.
Calculate and store the sample’s mean and standard deviation.
Repeat steps (1) and (2) 500 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.

samp_mean <- rep(NA, 500)
samp_sd <- rep(NA, 500)
n <- 60

Now we’re ready for the loop where we calculate the means and standard deviations of 500 random samples.

for(i in 1:500){
  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 500 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])

5.5 On your own

Using the function plot_ci() (which was downloaded with the data set), we are able to plot the first fifty 95% confidence intervals of our 500. What proportion of the 50 plotted confidence intervals include the true population mean? Is this proportion exactly equal to the confidence level? If not, explain why. Devise and implement a process to calculate the number (and proportion) of confidence intervals that include the true population for all 500 95% confidence intervals - you don’t want to have to plot and count by hand.
```
plot_ci(lower_vector, upper_vector, mean(population))
```
Suppose we want 90% confidence intervals instead of 95% confidence intervals. What is the appropriate critical value?
Construct 500 90% confidence intervals. 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_ci function, plot the first 50 90% confidence 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? Using the method which you implemented in question 1 of the On Your Own section, determine the number (and proportion) of the 500 randomly generated 90% confidence intervals that include the true population mean.