There are three kinds
of lies: Lies, Damn Lies, and Statistics.

Mark Twain

Chapter 9

Sec 9.2

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.