Topic 19 Central Limit Theorem

Learning Goals

  • Use the Central Limit Theorem to construct confidence intervals
  • Use confidence intervals to answer research questions
  • Make quantitative predictions about how changing sample size





Discussion

In the video/slides, we saw

Exercises

A template RMarkdown document that you can start from is available here.

Exercise 1

We know from the Central Limit Theorem (CLT) that at large enough sample sizes (roughly \(n \geq 30\)):

\[ \hat\beta \sim N(\beta, \hbox{SE}^2) \]

  1. Given a property of the normal distribution, complete the following probability statement:

\[ P(\hbox{???} < \hat\beta < \hbox{???}) = 0.95 \]

  1. Rearrange the probability statement above so that it looks like:

\[ P(\hbox{something with } \hat\beta \hbox{ and } SE < \beta < \hbox{something with } \hat\beta \hbox{ and } SE ) = 0.95 \]

  1. Explain how your work in (b) shows how we can use coefficient estimates and their standard errors to construct a 95% confidence interval.

  2. Would a 68% confidence interval be narrower or wider than a 95% confidence interval? What about a 99.7% confidence interval?



Note: Technically, in linear regression, the sampling distribution of the coefficients follows the T distribution, but at typical sample sizes where the CLT applies (\(n \geq 30\)), the T distribution is nearly indistinguishable from the normal distribution.



Exercise 2

Based on the CLT, we know that standard error has a certain relationship with sample size.

Suppose that I wanted the standard error of my estimate to be \(A\) times smaller. How would my sample size have to change to achieve this?



Interlude: Interpreting confidence intervals

Our work in Exercise 1 shows that we can create 95% confidence intervals with:

\[ \hbox{estimate} \pm 2\times\hbox{std. error} \]

That is, we can interpret the probability statement below:

\[ P\left(\hat\beta-2SE < \beta < \hat\beta+2SE \right) = 0.95 \]

as saying: the probability that a 95% confidence interval from a random sample contains the true population value is 95%.

  • The correct interpretation:
    95% of all possible samples will produce 95% CI’s that cover the true value. The other 5% are based on unlucky samples that produce unusually low or high estimates.
  • The INCORRECT interpretation:
    We cannot say that “there’s a 95% chance that the true population value is in the 95% CI from this particular sample.” Technically, the population value is either in the interval or it’s not, so the probability is simply 1 or 0.
    e.g., If the true population value is \(\beta = 1\), and my CI is (2,4), there is a 0% probability that the truth is in my interval. If my CI were (0.5,3.5), there is a 100% probability that the truth is in this interval.

Note that this also applies to bootstrap confidence intervals, and 95% is called the coverage probability or confidence level.



Exercise 3

In addition to using the reported estimate and standard error from our model output, we can use a handy function in R called confint() to compute confidence intervals.

If we fit a model called mod, we can use confint(mod, level = 0.95) to obtain 95% confidence intervals.

We’ll practice with a dataset containing information on house prices in upstate New York.

library(readr)
library(dplyr)
library(ggplot2)

homes <- read_tsv("http://sites.williams.edu/rdeveaux/files/2014/09/Saratoga.txt")
# Create an indicator of whether or not a house has a fireplace
homes <- homes %>%
    mutate(HasFireplace = Fireplaces > 0)

  1. Do we have evidence for a real, meaningful effect of square footage (Living.Area) on Price for houses with a fixed Age? A meaningful effect is at least 10 dollars per square foot. Interpret the relevant coefficient, and provide a 99% confidence interval to support your answer.

  2. Does your confidence interval contain the true population value of the relevant regression coefficient?

  3. Repeat parts (a) and (b) for the research question: Do we have evidence for a real, meaningful effect of Age on the chance of a home having a fireplace at fixed square footages? A meaningful effect is one with an odds ratio of at least 1.1 if the effect is positive or at most 0.9 is negative.