STAT 280 Lab 6: Sampling Distributions of Statistics

Preliminaries

Access the following site: http://www.stat.rice.edu/~hflores/stat280_gmdavis/
Select stat280L6.xls (you may also right-click and "save-as" to save it to your user account), to be used in Objectives 3&4.

Objectives

This lab will serve as a "complete" review of Chapter 5 of your book.  By the end of the lab you should be motivated to think more deeply about: 
•	What a statistic is (to refresh your memory)
•	What a sampling distribution of a statistic is
•	The definition of the sampling distribution of the sample count and sample proportion (in a Binomial random variables) and applying it
•	The definition of the sampling distribution of the sample mean (for any random variables) and applying it

Setup

Understand now that we are moving back into the "realm of statistics".  In reference to our previous labs, we will now have a known sample and an unknown population.  Yet, at times we will make probability calculations AS IF we knew the population.  Underlying this is the following idea:  If I believe that a random sample taken is "representative" of the unknown population, then I will proceed to make probability calculations on the sample as if it were my population in order to form conclusions that generalize to the unknown population.  Consequently, since the sample IS NOT my population we should expect some measure of sampling error which we will be able to compute later.  This idea will be represented mostly in Objectives 3 & 4.

Complete all of the following exercises.

Objective 1: Statistics?

- Define the term "statistic" with the technical definition AND in your own words.
- Which of the following are statistics? 
{sample mean, population mean, sample standard deviation, sample median, population median, the sample itself, the ordered sample}

NOTE: We are often tempted to say "mean", but we should be careful about defining which we are referring to {sample or population}.  Some terms might seem to be indicative of the sample like "average" or "count" but it should be clear to you by now that this is still not good enough.  Let's try to be specific!

Objective 2: Sampling distributions of statistics

- Define the compound term "sampling distribution", technically and in your own words.
- Explain my error in thinking about sampling distributions in the following monologue:

	"So I've got my data here, it's just a column of 100 numbers.  I can make a histogram of the numbers...it looks kind of bell-shaped.  Anyway, statistics are supposed to have a sampling distribution, right?  Well, I computed the sample mean, it was 45.2.  So the sampling distribution of the mean is constant, and is always equal to 45.2.  As such, the sampling distribution of the sample mean can be represented as a horizontal line."

- Explain how the histogram of the sample is a sampling distribution.  In other words, explain how the entire sample has a sampling distribution.
- Is it possible for population parameters (i.e. population mean, population mode, etc.) to have sampling distributions?  Explain.
- (Intuitively) Why should I care about the sampling distribution of a statistic?

Objective 3: Sampling distributions in the Binomial case

- Define as clearly as possible what it means for a random variable to be distributed as "binomial".  What is the difference between the count and proportion?
- Give an example of a binomial experiment (you may use substitute 2 examples in your text for one of your own) and explain clearly how this satisfies "The Binomial Setting" on pg. 368.

Now open the assigned data file (see Preliminaries).  You should see 3 columns, labeled Experiments 1,2,3.  Here's what I did to get the data:

Experiment 1: I tossed a fair coin 50 times and wrote down the number of heads.  I did this a total of 100 times.
Experiment 2: I tossed a fair coin 1000 times and wrote down the number of heads.  I did this a total of 100 times.
Experiment 3: I tossed a fair coin 50 times and wrote down the number of heads.  I did this a total of 10000 times.

- Compute the average of the sample proportions for each experiment.  Explain how the accuracy of the average was affected by the differences between the 3 experiments.

But what about sampling distributions you ask?  Glad you asked...

- Define the sampling distribution for the sample count of each of these 3 experiments (hint: they're all binomial(n,p), just define what n and p are).
- Define (or explain your idea of) the sampling distribution for the sample proportion of each of these 3 experiments.
- Define the normal approximations for the sampling distributions of the sample count and sample proportion [BONUS: Plot a histogram of the data in all three experiments to see how the counts are approximated normally]
- Using the normal approximations, compute the following:
P[# heads <= 45], P[0.45 < proportion < 0.55].  How can we justify computing these probabilities for population parameters?

Objective 4: Sampling distributions in the "general" case

- Define the sample mean.  It is a statistic, so define its sampling distribution.
- Compute the following in experiment 3:
P[sample mean < 22], P[20<sample mean<30].  Why might I be interested in these calculations?

A Note on the Central Limit Theorem
-----------------------------------------------------
This theorem gives us a very deep and rich result: Given a simple random sample, the sample mean will have a normally distributed sampling distribution regardless of what distribution the data came from as the sample size increases.  THIS IS NOT to say that the histogram of my data will look bell-shaped eventually if I had enough data, but rather the histogram of the sample mean (given enough repeated identical experiments) will look bell-shaped.  Keep this result in mind as you move into Chapter 6, which will focus on making inferences involving the sample mean, whose distribution is determined for any "large enough" sample.