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.
## 800 1228 1598 1608 1955 2698
This is a normal distribution (although it’s hard to tell visually). The median (“typical”) size within the sample is 1598; mean would also be a good representation as this is a normal distribution
- Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?
It is very unlikely that another student’s distribution would be completely identical (I could give you the probability, but that was last week’s assingment). It could be similar, as it is a sample of the same population, but could also be different as it’s a small sample size (unlikely however).
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] 1479.403 1735.597This 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?
The sample mean must be normally distributed, and n must be greater than 0.
Confidence levels
- What does “95% confidence” mean? If you’re not sure, see Section 4.2.2.
It means ‘we are 95% sure that the mean of the populaton falls within this range.’
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?
The true population mean does fall within the CI. I would estimate the probability of my neighbor’s capture value falling within that range at 95%.
- 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.
I would expect 95% of them to capture the true population mean. This is becuase the CI is an estimate of how sure we are that the mean will fall within the CI.
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.
set.seed("1234567890")
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] 1479.403 1735.597On 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.
94% of the CI’s above contain the true population mean. This isn’t exactly equal to the confidence level, but it is close. With the amount of CI’s above, it’s actually as close as it can get; 95% would mean we wou;d need 47.5 of them to contain the mean. Small devations beyond what is shown above may also be due to chance.Pick a confidence level of your choosing, provided it is not 95%. What is the appropriate critical value?
The critical value for a confidence level of 99% is 2.58
- 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?
#I stole your code above for this
lower_vector <- samp_mean - 2.58 * samp_sd / sqrt(n)
upper_vector <- samp_mean + 2.58 * samp_sd / sqrt(n)
plot_ci(lower_vector, upper_vector, mean(population))
98% of the CIs contain the population mean. This is slighlty lower than the confience level, but as close as this set can get as there are only 50 samples. These CIs include more of the means than did CIs with a confidence level of 95%.