Sampling Distribution for the Sample Proportion

Qualitative Responses Thus far we have discussed quantitative data –The survey question we ask has required a numerical response How far do you live from campus? How long did the battery last? How much did you spend on health insurance? Etc. Now we shall ask questions that involve a qualitative response What is your political party? What color is your car? Are you a male?

Proportions as a Random Variable When we isolate one of the possible responses we can determine the proportion that gave that particular response The proportion of Democrats in Orange County The proportion of red cars in a parking lot The proportion of males at a football game If we are going to take a sample of size n Orange County voters, then before the sample is taken, the proportion of Democrats we will get in the survey is a random variable If we let X = the number of the n respondents that say they are Democrats (also a random variable since before the sample is taken we do not know the value of X), then the proportion of Democrats is

The Random Variable X X = the number of “successes” in n tries –Binomial; but because n = large -- can be approximated by a normal distribution –Mean  X = np –Variance = np(1-p)

_ THE RANDOM VARIABLE P Distributed approximately normal with:

Example Suppose 40% of the voters in Orange County are Democrats. What is the probability that in a sample of 400 randomly selected Orange County voters, there will be more than 180 (45%) Democrats?

What is the probability more than 45% of the sample will be Democrats? Z 2.04 ( )/ =.0205  =

Review –Normal distribution –Mean  = –Standard Deviation =