**Distributome Activity on Sample Sizes and the Accuracy of Polls**

## Introduction

Surveys about public opinions on controversial social issues are becoming increasingly frequent as topics such as the legalization of marijuana, abortion policy, marriage rights for homosexuals, and immigration policy are hotly debated in the media.

For example, both the Opinion Research Corporation (polling for CNN) and the Pew Research Center for The People and The Press conducted surveys of American adults in spring of 2011 to estimate the percentage of the public that favors the legalization of marijuana. The sample sizes in the two polls were 824 for the Opinion Research Corporation poll and 1504 for the Pew poll.

## Goals

This activity illustrates the inter-distribution relationships between Cauchy, Student’s T and Standard Normal (Gaussian) distributions. These relationships are used to provide a better understanding of how strongly sample sizes are related to the accuracy of polls.

## Hands-on Activity

In this activity you may assume that both of these pollsters use similar techniques that involve telephone interviews and weighting the answers given by individuals to align the respondents demographics with population values and finally averaging to produce unbiased and essentially normally distributed estimates. Below are 4 related, but complementary, problems regarding this study.

**Note**: The problems below may be appropriate for an undergraduate course in probability. The last part (Problem 4) would be more appropriate for masters level course (and should have a General Cauchy distribution tag and a tag for its relationship to the bivariate normal).

## Specific Problems and their Solutions

### Problem 1: Difference in Poll Accuracies?

The Pew poll had almost twice the sample size of the Opinion Research Corporation poll. What is the chance that it was more accurate than that poll for estimating *p = the percentage of American adults that favored the legalization of marijuana in spring, 2011*? Be sure to clearly define how you are interpreting “more accurate.” Also state and justify any assumptions you make in solving for this probability.

## Alternative approaches

### Alternative 1: Ratio of bivariate Normal variables.

See a Solution to First Problem: Alternative 1

### Alternative 2: Direct calculation of the marginal distribution

See a Solution to First Problem: Alternative 2

## Problem 2: Pooling Data across Polls?

Describe how you would combine the data from these two polls to form a single estimate of \(p\).

The obvious choice is to propose a linear combination of the two estimates weighting inversely proportional to the variances to get the smallest overall variance amongst such linear combinations.

## Problem 3: Are these probability estimates correlated?

What is the correlation between your estimate above and the individual estimate produced by the Pew poll?

Note that \(Cov(\hat{p},\hat{p_1})=Cov(\frac{1504}{2328}\hat{p_1}+\frac{824}{2328}\hat{p_2}, \hat{p_1})=\frac{1504}{2328}\sigma_1^2=\frac{\sigma^2}{2328}\).

## Problem 4: Accuracy of probability estimates?

What is the probability that your combined estimate (from the second problem) is more accurate than the estimate based only on the Pew poll?

We want \(P[|\hat{p}-p| < |\hat{p_1}-p| ]\).

## Generalized Cauchy distribution CDF derivation

To derive the generalized Cauchy distribution CDF directly, we start with the bivariate normal distribution of ”X” and ”Y”:

See the Derivation of the Cauchy CDF

## Cauchy, Student’s T and Gaussian distribution interrelations

The Student’s T-distribution represents a one-parameter homotopy path connecting Cauchy and Gaussian Distribution:

- \(Cauchy=T_{(df=1)} \longrightarrow T_{(df)}\longrightarrow N(0,1)=T_{(df=\infty)}\).
- See the Distributome Navigator, Cauchy, Student’s T and Gaussian distribution calculators.
- Explore the relations between these and many other probability distributions using the interactive graphical Distributome Navigator.

## Conclusions

Increasing the sample size may help significantly in certain situations – but not as much as intuition often suggests.