Confidence Intervals
Intro to OLS Regression
POLS 3316: Statistics for Political Scientists

Tom Hanna

2023-11-12

Confidence Intervals

  • Review \(X^2\) and t-test
  • General hypothesis testing: scores and critical values
  • Confidence intervals

\(X^2\) and t-test

  • Note on formulas: Numbers in numerator or denominator of a fraction all get solved together as if in parentheses. PEMDAS is shorthand, but the P should really be G for “grouping symbols” and includes: parentheses ( ), brackets [ ], braces { }, and even fraction bars.

Terms

  • null hypothesis
  • alternative hypothesis
  • critical value (cutoff point)
  • critical value is found using degrees of freedom and desired probability or alpha. (In z-scores, only probability is needed.)
  • probability or p - confidence level

Test Scores and Critical Values

  • if the test score is greater than the critical value, reject the null

  • The critical value is minimum

  • Above that value, the probability that the null hypothesis is true is equal to or less than the target so we reject

  • This all leads to confidence intervals

# Confidence intervals

## What is a confidence interval?

  • Range where we could expect to find the true measurement based on the probability level we specify if we had the full population data instead of just the sample.
  • Range where we expect to find the true population parameter given the sample statistic
  • If we say the 95% confidence interval of the mean is from 45 to 55, it means we are 95% certain that the true population mean lies somewhere in that range.

Getting the confidence interval

  • The confidence intervals are based on the thing we actually measured, the probability level, and the distribution.
  • If we know the population parameters (especially standard deviation) we can create simple CI using 68-95-99 rule or accurate CI using z-scores
  • For small samples or uncertain population standard deviation we use appropriate distributions, such as Student’s t, to adjust for both these uncertainties

Using t-scores to find confidence intervals

  • For small samples or uncertain population standard deviation we use t-scores to adjust for these uncertainties

              - If we wanted to get a 95% confidence interval with 68-95-99, we would look for a value 2 standard deviations either side of the statistic we want to find a CI for 
          - With t-test scores we divide the standard deviation by the square root of sample size and multiply times a t-table critical value instead.
          - We can get the critical value using the qt() function in R as in the examples

Using t-scores to find confidence intervals

  • \(CI_\mu\) Confidence interval of the population mean

  • \(CI_\mu\) = \(\bar{x} \pm t \frac{\sigma_{s}}{\sqrt{n}}\)

Where:

  • \(\bar{x}\) = Sample mean
  • \(\mu\) = Population mean
  • \(\sigma_{s}\) = sample standard deviation
  • n = sample size

What about the normal distribution?

The 68-95-99 rule gives an oversimplified CI.

For normal distributions, the actual formula is:

What about the normal distribution?

\(CI_\mu\) Confidence interval of the population mean

\(CI_\mu = \bar{x} \pm z \frac{\sigma}{\sqrt{n}}\)

Where:

\(\bar{x}\) Sample mean \(\sigma\) population standard deviation \(n\) is the sample size

  • We can get the z-score from a z-table or from the qnorm() function in R

Examples:

We’ll go through a couple of example computations, then I’ll provide you some sample R lab scripts (Quarto files).

Examples:

  • Sample mean: 50 = \(\bar{x}\)
  • Sample standard deviation 5 = \(\sigma_{s}\)
  • Sample size 18 = \(n\)
  • Looking for a 95% confidence interval.

We need the t-score for a two-tailed test:

Code 
qt(.025,17)
[1] -2.109816

Examples:

Code 
qt(.025,17)
[1] -2.109816

Then we can fill in the formula:

  • \(CI_\mu\) = \(\bar{x} \pm t \frac{\sigma_{s}}{\sqrt{n}}\)
Code 
50 + c(-2.11, 2.11) * 5 / sqrt(18)
[1] 47.51334 52.48666

Example 2:

Using the z-score:

  • Sample mean: 50

  • Population standard deviation: 5

  • Sample size: 500

  • We need a z-score for 95% (.025% above and .025% below):

Code 
qnorm(.025)
[1] -1.959964

Example 2

Then we apply the formula:

\(CI_\mu = \bar{x} \pm z \frac{\sigma}{\sqrt{n}}\)

Code 
50 + c(-1.96, 1.96) * 5 / sqrt(500)
[1] 49.56173 50.43827

Is this just about means?

NO!

The biggest use I see of confidence intervals is confidence intervals for regression coefficients

  • \(y = \alpha + \beta X + \epsilon\)

If we find that the \(\beta\) in the equation defining the relationship between our explanatory (x) and dependent (y) variables is 0.9, the confidence interval can tell us that it’s actually 95% likely that the true relationship is between 0.88 and 0.92.

Even better, we can graph it…

Sample of graphing confidence interval in OLS

Additional Resources for Confidence Intervals

https://www.statisticshowto.com/probability-and-statistics/confidence-interval/#WhatisCI

https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/confidence-level/

https://www.statisticshowto.com/95-percent-confidence-interval/

## Authorship, License, Credits

Creative Commons License

POLS3316, Fall 2023, Instructor: Tom Hanna