Sample Proportions and Point Estimation

Sample Proportions

Let p be the proportion of successes of a sample from a population whose total proportion of successes is p and let mp be the mean of p and sp be its standard deviation.

Then

 The Central Limit Theorem For Proportions mp = p  For n large, p is approximately normal.

Example

Consider the next census.  Suppose we are interested in the proportion of Americans that are below the poverty level.  Instead of attempting to find all Americans, Congress has proposed to perform statistical sampling.  We can concentrate on 10,000 randomly selected people from 1000 locations.  We can determine the proportion of people below the poverty level in each of these regions.  Suppose this proportion is .08.  Then the mean for the sampling distribution is

mp = 0.8

and the standard deviation is

Point Estimations

A Point Estimate is a statistic that gives a plausible estimate for the value in question.

Example

x is a point estimate for m

s is a point estimate for s

A point estimate is unbiased if its mean represents the value that it is estimating.