Transcript Confidence Intervals - University of Notre Dame

The Guessing Game
The entire business of Statistics is dedicated to
the purpose of trying to guess the value of
some population parameter. In what follows
the population parameter (the target
parameter) will be either a mean or a
proportion p.
All that we have at our disposal is n values
taken at random from the population
x1, x2, …, xn
and their average and their variance:
Not much to go on, but when n is large
(admittedly we don’t know what that means,
but n ≥ 30 will make us feel better) we have
the wonderful
Central Limit Theorem
that tells us that the distribution of the
sample mean (the average)
is
approximately normal
x
Trouble is, we don’t know
and we don’t
know
. We just have the following picture:
x
and the one number we know, is
somewhere on the horizontal line. Actually
“somewhere” is a cop-out, we know exactly
where is, we just don’t know where the
red curve is relative to ! Could be like
this
x
x
or like this:
(in both figures the blue dot is
x
)
Looks somewhat hopeless, but there are
some statements we can make for sure.
The next slides show four of them!
Hopefully you’ll catch on. Replace the
blue dot (that represents
) with the RED
DOT (that represents the standardization
of
)
x
x
80% chance the red dot is inside the
blue bars. (why?)
85% chance the red dot is inside the
blue bars. (why?)
90% chance the red dot is inside the
blue bars. (why?)
95% chance the red dot is inside the
blue bars. (why?)
The Theory
1.
2.
3.
4.
What went on in each of the previous four
slides?
Let’s see. We picked a percentage of area
80% - first slide
85% - second
90% - third
95% - fourth
From the chosen percentage we got
(via the standard normal tables)
1.
2.
3.
4.
symmetric z-scores
80% gave us -1.285
1.285
85% gave us -1.439
1.439
90% gave us -1.649
1.649
95% gave us -1.960
1.960
In fact, if you give me any
positive area ≤ 1 (call it 1 I can find the corresponding
symmetric z-scores
by looking for the area value
)
(figure it out from the figure!)
The two z-scores you get are written as
±Z
/2
and the number 1 is called
confidence coefficient if in decimal form
confidence level
if in percent form
Confidence Intervals
Why are we using the word “confidence”?
Confidence in what?
Of course, we hope it is confidence in our
prediction! In fact we want the confidence
level to be just the probability that our
prediction is correct.
Trouble is ….
We haven’t predicted anything !!
We just have established that
For any confidence coefficient 1 find z-values
and
we can
so that
P(red dot between
and
)=1-
Recall that the red dot stands for the
standardized value
So we obtain the statement
that a little 7-th grade algebra transforms into
This is translated into English as:
is inside the interval
… with probability 1 Or, in slightly different (and more pompous
sounding) words
“We are (1 - )% confident that
is inside
the interval
We call this interval the (1 -
)%
confidence interval.
THAT’S OUR PREDICTION !
One last step: what do we use for sigma ?
If we know it (sometimes we do) …
HALLELUYA !
If not, we approximate sigma using the
sample standard deviation
where s is the (computed) sample
standard deviation.
An example
The numbers shown have been obtained
as time T (in seconds) elapsed from the
time the cage door is opened to the time of
exit from the cage for 40 lab mice in
separate cages; (20 of the mice have been
given a tranquilizer, the other 20 a
placebo, but this is for another problem
later.)
Construct the following
confidence intervals
for the mean of T
The Question:
1. 90% confidence interval
2. 95% confidence interval
3. 99% confidence interval
4. 30% confidence interval
5. 10% confidence interval
The data:
3.5 2.2 1.4 3.6 3.5 2.6 2.7 2.1 1.9 4.1
2.7 2.8 2.3 1.9 1.3 3.3 2.8 2.6 2.1 3.8
4.3 4.4 2.8 2.0 3.3 4.1 1.4 3.1 2.8 3.0
4.1 4.2 3.8 3.9 4.1 3.4 3.1 1.3 4.5 3.2
The sample mean and standard deviation
are:
therefore
The z-scores
For each
we compute z’s (from my
“stats” program or from the table)
1. 90%
1.645
2. 95%
1.960
3. 99%
2.579
4. 30%
0.385
5. 10%
0.126
The answers
Using the formula
we get the intervals
1.
2.
3.
4.
5.
90%
95%
99%
30%
10%
[2.760, 3.239]
[2.715, 3.285]
[2.625, 3.375]
[2.944, 3.056]
[2.982, 3.018]
Note that the higher the confidence the wider
the interval. Is this reasonable?
or
who’s gonna win the elections?
Quite often one needs to estimate what
proportion p of a population prefers
option A over option B.
One takes a “large enough” random
sample of the population, counts how
many prefer A, divides by the size n of
the sample and gets a number,
denoted by
(a statistic!) .
Of course
is a random variable, and it turns
out that it is an unbiased estimator of p, that is
E( ) = p
If we knew the standard deviation
of
we could construct confidence intervals for p as
we did for the parameter
.
(If n is big enogh the Central Limit Theorem still
holds)
We can show that
= pq/n (remember
that q = 1 - p), but this is tautological (we don’t
know p !)
However, if n is large enough, we can use
for p and proceed as with
.
In other words, we use
instead of
and
s
instead of
x
n
and get the interval
An Example
What proportion p of Notre Dame students
know a language other than English?
In a random sample of 1,500 Notre Dame
students, 855 stated they knew some
language other than English.
Develop a 98% confidence interval for p
based on this sample.
We have
Therefore
= 0.02
= 2.33
(why?)
Now
= 855/1500 = 0.57 and therefore
we approximate
with 0.0128 (why?)
We get the 98% confidence interval as
(0.57 – 2.33x0.128, 0.57 + 2.33x0.128)
That is
(0.54, 0.60)
THE FORMULAS
• To estimate
with confidence 1 -
• To estimate
with confidence 1 -