Transcript Lecture covering Reading Handout #1 (3/31/05) -

STAT 1301
Expected Value, Standard Error
for
AVERAGES
RECALL:
Sampling without replacement from a
population is often analogous to
drawing with replacement from a box
Recall: We use sample statistics
to estimate population parameters
We will study 2 types of population
parameters in this course:
1. averages
2. percents
TODAY: We focus on AVERAGES
Chance Process

Take a random sample from a
population and find the Sample
Average

We use X (read “X-bar”)
to denote the Sample Average
Take a random sample of
size n from a population

Question:
– What do we “expect” the Sample
Average to be?

Answer:
– We expect it to be “close” to the
population average (average of the
box)
Terminology

The Population Average is called
the EXPECTED VALUE of the
Sample Average
– Notation:
EV(X) = population average
Figure 1
sample average

We DO NOT “expect” that X will be exactly
equal to the population average

More precisely, we state the situation as
follows:
X = Population Average + Chance Error
Figure 2
chance error
How large do we expect the
chance error to be?

The likely size of the chance error is
measured by the Standard Error of
the Sample Average
– Notation: SE(X)
Standard Error of X
SD of Population
SE(X) =
sample size
Note: As sample size increases,
SE(X) decreases .
The fact that SE(X) decreases
as sample size increases is
just common sense

larger samples tend to result in more
accurate estimates of the true
population average
(when we take a probability sample)
Summary
A sample average is likely to be
around its Expected Value (i.e. the
population average), but is likely to
be off by a chance error similar in
size to the Standard Error (i.e. SE( X).
Another Point

Approximately 95% of sample
averages will be within 2 SE(X) of the
population average

Nearly all sample averages are within
3 SE(X) of the population average