Foundations for statistical inference - Confidence intervals
Abdelmalek Hajjam
10/9/2019
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.
## [1] 1411.167The distribution of the sample is right skewed unimodal distribution. The typical size of my sample I wuld say would be 1500 - 2000 according to the histogram. I interpret ‘Typical’ for me o mean most common and most seen.
- Would you expect another student’s distribution to be identical to yours? Would you expect it to be similar? Why or why not?
No, because the sample size is so small, I would not expect the distribution of another student to be identical. But they could be similar since samples are taken from the same population.
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] 1310.320 1512.013This 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 samples have to be random with 10% condition when sampling without replacement. The sample size has to be 30 or greater. There should be no extreme skews.
Confidence levels
- What does “95% confidence” mean? If you’re not sure, see Section 4.2.2.
Means that we are 95% or more sure that, the True population mean, lies within this interval.
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?
Yes, my confidence interval (1435.747, 1705.419) captures this value 1499.69, that is the true average size of houses in Ames. Running a simulation shows that our interval capture this value, which leads into believing that my class neighbor interval will also capture this value.
- 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.
Because we are using a 95% confidence interval, 95% of the class will capture the True mean of the population.
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] 1469.242 1718.391On 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 get 46/50 = 92% of my confidence intervals capture the true population mean. Because the sample size are very small, Confidence intervals change during simulation. This portion is smaller than the confidence level.
- Pick a confidence level of your choosing, provided it is not 95%. What is the appropriate critical value?
I choose the confidence level of 99% confidence interval. In this case the appropriate cristical value 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?
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))
I see that 100% of the confidence intervals capture the true population mean. I was suppose to miss some as I used critical value of 2.58, i.e. 99% confidence level. i.e.99% is close to 100%. I think this is due to the intervals being very wide.
## [1] 1457.266 1862.135