Chapter 10– Estimating Voter Preferences

1. Chapter 10– Estimating Voter Preferences Statistics is the science of making decisions in the face of uncertainty. We use information gathered from a sample to draw conclusions about the characteristics of the population from which the sample was selected. Since we have incomplete information, our conclusions will have some uncertainty associated with them. The uncertainty is called sampling error – different samples selected by the same method will yield slightly different results.

2. Margin of Error • If we are interested in estimating a parameter associated with the population, the uncertainty manifests itself in the form of the margin of error of estimation. • Different methods of sampling will tend to yield different degrees of uncertainty, or different margins of error. • In any estimation situation, part of the responsibility of the statistician is to choose that sampling method that tends to yield the smallest margin of error.

3. Scenario – Predicting an Election Outcome • Jane Goodgal (age 72) is the Democratic candidate for mayor of Taxes, Texas, a small town with 480 registered voters. • You are employed by Winners Political Consulting Group to predict the outcome of the mayoral election, using a sample of 100 voters. • The population of voters is diverse, with three different ethnic groups – African-American, Caucasian, and Hispanic.

4. Possible Sampling Methods • There are different sampling methods that could be used. Specifically, you have the choice of using either simple random sampling or stratified random sampling. • If there is a stratification variable that is strongly related to the outcome variable, then stratified random sampling tends to yield a smaller margin of error, and should be used instead of simple random sampling.

5. Simple Random Sampling (without Replacement) • To choose a simple random sampling of size n, we use the following procedure: • i) Obtain a complete list of members of the population, • ii) Assign a unique ID number to each member of the population, • iii) Go to a table of random numbers, choose a starting point, and read numbers in that column, until n different ID numbers have been selected (ignored repeated numbers in the list).

6. Stratified Random Sampling • Suppose that the population of size N is partitioned into L strata, with the ith stratum having Ni elements in it. The (unknown) proportion of supporters in the ith stratum is Pi. • We want to choose a simple random sample from each stratum, so that the total sample from the population has fixed size n. We must choose at least two elements from each stratum.

7. If we knew Pi for each i = 1, 2, …, L, then we would choose stratum samples of sizes • Since we do not know the Pi’s (if we did, we would not need to do a poll), we must guess likely values, based on prior information, such as the conjectures by the Democratic Party chairperson.

8. Use of Prior Information Lillian White, the Democratic Party chairperson for the town, believes, based on past experience, that: • 75% or more of the African-American voters will support Ms. Goodgal, • 30% or less of the Caucasian voters will support her, • about 50% of the Hispanic voters will support her, • Support for Ms. Goodgal will be much stronger among voters over 50 years old, and • females will be slightly more supportive than males.

9. Parameter Estimation • The parameter to be estimated is P, the proportion of the population of voters who intend to vote for Ms. Goodgal for mayor. • A sample will be selected, and each person in the sample will be asked, “Do you intend to vote for Ms. Jane Goodgal for mayor.” Possible responses to the question are “Yes”, “No”, or “Uncertain”.

10. Estimation by Simple Random Sampling • If sampling without replacement is used, the point estimate of P is , where x is the number of supporters in the sample. • The margin of error is then . • Then a 95% confidence interval estimate for P is given by

11. Estimation by Stratified Random Sampling • If we use stratified random sampling (without replacement), the point estimator of P is • with margin of error given by • The 95% C. I. for P is then

12. Implementing the Poll • A list of the registered voters in the population is given in Table 10.1 on page 107, together with some (possibly) relevant information about each person. • You should select one of the two sampling methods, and select your sample. I will provide a table of random numbers. • I will then give you the responses of the selected persons to the question. • You should then answer the questions in Assignment: Analyzing the Data.