Heights  Put your height in inches on the front board.  We will randomly choose 5 students at a time to look at the average of the heights in this class. Look at averages of 20 samples from the class. Is this a statistic or parameter? Create a histogram of the averages you found.  What is the average height of the whole class? Is this a statistic or parameter? How does it compare to the center of your histogram?

Section 7.1 Second Day Sampling Distributions

Recall…  What is the difference between a parameter and a statistic?  Which symbol do I use for the following?  Population proportion?  Sample mean?  Population mean?  Sample standard deviation?  Sample proportion?  Population standard deviation?

Sampling Distributions  A sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the population.

Describing Sampling Distributions  We can use the tools of data analysis to describe any distribution, including a sampling distribution.

Example  Let the population be all the numbers on a die. Roll two dice (an SRS of two) and find the average value for each set of rolls. Complete this 20 times and make a histogram. Describe the histogram.  Can we construct the ACTUAL probability distribution? List all possible outcomes and their x-bars. If we find the mean of the x-bars, then this is the actual mean.

So…

Bias  So far we’ve talked about the bias of a sampling method. However, it is often useful to talk about the bias of statistic. When talking about a statistic, bias concerns the center of the sampling distribution.

The bias of a statistic  When a sampling distribution centers around the true value of the parameter, we say it is unbiased.  In other words, a statistic is unbiased if the mean of its sampling distribution equals the true value of the parameter being estimated.  There is no SYSTEMATIC tendency to under- or overestimate the value of the parameter.

The variability of a statistic  The variability of a statistic is described by the spread of its sampling distribution.  LARGER SAMPLES GIVE SMALLER SPREAD.  Notice that this is saying that larger samples are good. However, it says NOTHING about the size of the population.  Q: Does the size of the population matter? A: No. We usually require the population be ten times larger than the sample. So if n = 20, population should be at least 200.

A Note about Population Size  Example: Suppose the Mars Company wants to check that their M&Ms are coming out properly (i.e. Not broken, not undersized, etc.).  It doesn’t matter if you select a random scoop from a truckload or a large bin. (Meaning: population size doesn’t matter)  As long as the scoop is selecting a random, well-mixed sample, we’ll get a good picture of the quality of M&Ms.

Bias vs. Variability What does the bull’s-eye represent? What do the darts represent?