Sampling and Estimating Population Percentages: Chance Error, Bias, and Standard Errors, Study notes of Statistics

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STAT 2

Lecture 25:

Chance errors in sampling

Recall: sampling ● It's usually impractical to survey a whole population, so we take a sample instead ● A well-chosen sample of a few thousand can give a accurate picture of the whole US population

Recall: bias ● Selection bias means the method for selecting the sample is more likely to select some members of the population than others in a way that's not anticipated ● Non-response bias occurs because those who don't respond differ from those who do

Recall: sampling methods ● Quota sampling lets interviewers select subjects according to some quotas; strong risk of bias ● Simple random sampling selects members of a population with equal probability, and is unbiased ● In practice, multistage cluster sampling is often employed

I

A chance model for simple random sampling

The problem ● We wish to estimate the percentage of a certain population with a certain characteristic e.g. the percentage of registered American voters who would vote for Obama

Estimating parameters from statistics ● For a simple random sample, the sample percentage is our estimate of the population percentage ● For a more complex sampling scheme, we may need to adjust our sample percentage for possible biases

The box model for a simple random sample ● Need one ticket for each member of the population ● Write a “1” on the ticket if they have the characteristic; write a “0” if they don't ● Put the tickets in a box, shuffle, and draw n tickets without replacement

Box SD ● The SD of the box of zeroes and ones, according to the two-valued box SD formula, is Box SD = sqrt(prop. of zeroes times prop. of ones)

Standard error of the sum of draws ● If the population size is huge, there will be no real difference between drawing with and without replacement ● SE of sum of draws (with replacement) = sqrt( n )*(box SD) = sqrt( n * prop. ones * prop. zeros)

Example ● Suppose the true percentage of voters who support Obama is 50%. ● The expected percentage of a sample who support Obama is 50% (no matter what the sample size is).

Example ● If n =100, the standard error of the sample percentage is sqrt(0.50.5/100)100% = 5% ● If n =1000, SE = 1.6% ● If n =10000, SE = 0.5% The SE of a percentage goes down like the square root of n. (The SE of a sum goes up like sqrt( n ).)

II

Sample percentages and the normal distribution

The distribution of sample percentages ● The percentage of 1's among draws from a box of 0's and 1's is likely to be around the expected value (of the percentage), give or take its SE ● SE of percentage depends on percentage of 1's, number of draws, and whether they're with or without replacement