There are three kinds
of lies: Lies, Damn Lies, and Statistics.
A Gallup Poll found that 210 out of a random sample of 501 U.S. teens knew that Columbus discovered America in 1492. That is, the PROPORTION of U.S. teens knowing this fact was 210/501 or .42 which can be stated as 42% but we will leave the proportions as decimals.
Determining the center, shape, and spread of the sampling distribution (p hat) can be done by connecting proportions and counts.
Choose an SRS of size n from a large population with population proportion (p) having some characteristic of interest. Let (p hat) be the proportion of the SAMPLE having that characteristic. Then:
1) the MEAN of the sampling distribution is EXACTLY (p)
2) the standard deviation of the sampling distribution is square root of [(p(1-p)/n}
These facts must be memorized. You will be called on to use the recipe in #2 to determine missing standard deviations.
Because the mean of the sampling distribution of (p hat) is always equal to the parameter p, the sample proportion (p hat) is an UNBIASED ESTIMATOR of (p). The standard deviation of (p) hat gets smaller as the sample size n increases because n appears in the denominator of the formula for the standard deviation. That is, (p hat) is less variable in larger samples. The sample size n is under the square root sign, so to cut the standard deviation in half, we must take a sample four times as large, not just twice as large.
RULES OF THUMB:
1) Use the recipe for the standard deviation of p hat only when the population is at least 10 times as large as the sample
2) Use the normal approximation to the sampling distribution of p hat for values of n and p that satisfy np > 10 and n(1-p) > 10.
See Ex. 9.l7 and 9.8 pages 507-509 for more detail.