8-1 Estimation
Estimating μ when σ is Known
Determine some z scores
Determine the z score so that 70% of any data
set will fall between z and –z.
Determine the z score so that 75% of any data
set will fall between z and –z.
Determine the z score so that 80% of any data
set will fall between z and –z.
Determine the z score so that 90% of any data
set will fall between z and –z.
Determine the z score so that 99% of any data
set will fall between z and –z.
What use is this?
Confidence – being relatively sure that
data falls in a certain area.
When  and s are known, the idea here
is to be reasonably* sure that μ is
nearby.
* reasonably sure tends to be a matter of opinion,
but we will generally define it later
Assumptions
Simple random sample, size n, from population
of x values
σ is known
If x distribution is normal, this method works
If x distribution is unknown, n ≥ 30
If x distribution is skewed or not mound
shaped, n will need to be even higher.
 is called the point estimate of μ
Margin of error is | ― μ| (magnitude of  ―
μ)
Confidence Level
The reliability of the estimate
The confidence interval is the area
where you can state with a certain
level of confidence (c) that the mean
lies.
There will be critical values so that the
area from –zc to zc will be equal to c
This will look confusing
But use logic…
This will look confusing
But use logic…
x μ
zc 
σ
n
zc  σ
n
 x μ
This will look confusing
But use logic…
x μ
zc 
σ
n
x μ
zc 
σ
n
zc  σ
z c  σ
n
n
 x μ
 x μ
This will look confusing
But use logic…
x μ
zc 
σ
n
x μ
zc 
σ
n
zc  σ
z c  σ
Remember also that zc and –zc are the boundaries
… therefore
n
n
 x μ
 x μ
Therefore
If –zc and zc are the boundaries, then we
can rewrite the equation to be


σ
σ
P  z c 
 x  μ  zc 
c
n
n

Therefore
If –zc and zc are the boundaries, then we
can rewrite the equation to be


σ
σ
P  z c 
 x  μ  zc 
c
n
n

And we can call the margin of error
E  zc  σ
n
Ergo
P(E  x  μ  E)  c
And a little algebra lets us turn this into
c is your confidence
level – 95%, 93%,
whatever…
Ergo
P(E  x  μ  E)  c
And a little algebra lets us turn this into
c is your confidence
level – 95%, 93%,
whatever…
x E  μ  x E
Practice problem
In the third week of July, a random sample of
40 farming regions gave a sample mean of
 $6.88 per 100 lbs of watermelon. Assume
that σ is known to be $1.92 per 100 lbs.
A) Find the 90% confidence interval for the
population mean price per 100 lbs that farmers in
this region get for their watermelon crop. What is E?
B) A farm brings 15 tons of watermelon to the market.
Find a 90% confidence interval for the population
mean cash value of this crop. What is the margin of
error?
Calculator Slide
As usual, some by hand….
But when you need to use the calculator
STAT: TESTS 7: Zinterval
Now if you chose Stats then you will toggle
down and type in the information you have.
If you choose Data, then you have to refer to
the list (but still put in σ).
C level refers to confidence level