Statistics are no substitute for judgment.
Sample proportions arise most often when we are interested in categorical variables like "What proportion of U.S. adults have watched Survivor II?" or What percent of adult s attended church last week?"
When we record quantitative variables, the income of a household, the lifetime of a car brake pad, we are interested in other statistics, such as the median or mean or standard deviation of the variable. Sample means are just averages of observations, they are among the MOST COMMON statistics.
Let's agree that it makes sense that:
1) Averages are less variable than individual observations.
2) Averages are more normal than individual observations making the use of sample means a frequent choice for use in statistical inference.
THE SAMPLING DISTRIBUTION OF `x is the distribution of `x in ALL POSSIBLE samples of the same size from the population
MEAN AND STANDARD DEVIATION OF A SAMPLE MEAN:
Suppose that `x is the mean of an SRS of size n drawn from a large population with mean µ and standard deviation δ. Then the mean of the sampling distribution of `x is µx = µ and its standard deviation is
δx = δ/square root of n.
Whatever the shape of the population distribution these facts are true:
1) The sample mean `x is an unbiased estimator of the population mean µ
2) The values of `x are less spread out for larger samples.
3) You should only use the above for δx when the population is at least 10 times as large as the sample.
CENTRAL LIMIT THEOREM:
Draw any SRS of size n from any population whatsoever with mean µ and finite standard deviation.
When "n" is LARGE, the sample distribution of the sample mean `x is close to the normal distribution N(µ, /sqrt n) with mean µ and standard deviation /sqrt n.
VERY IMPORTANT: THE CENTRAL LIMIT THEOREM ALLOW US TO USE NORMAL PROBABILITY CALCULATIONS TO ANSWER QUESTIONS ABOUT SAMPLE MEANS FROM MANY OBSERVATIONS EVEN WHEN THE POPULATION DISTRIBUTION IS NOT NORMAL.
How large a sample size is needed for its mean to be close to normal depends on the population distribution and more observations are required IF the shape is far from normal. The BIG IDEA of a sampling distribution is that if we keep taking random samples of size n from a population with mean µ and find the sample mean for each sample, collect all the sample means and display their distribution, we have a sampling distribution.
The Chapter Review on page 527 does a good job of summarizing the concepts in Chapter 9. Check it out!
Note: Exponential distributions are special density curves used to describe the lifetime in service of electronic components and the time required to serve a customer or repair a machine.