Transcript Section 9.1 Second Day

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…
x  
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?