Math 1080 Lecture 21 Confidence Interval for Variance and Standard Deviation An Example

Today I'll discuss confidence interval
for sigma and sigma squared--

that's standard deviation and variance--
using the chi-squared distribution.

Now, these are to be estimated (sigma
and sigma squared) for the population.

Remember, population
is what you're studying.

And sample is what
you have in hand.

You don't have access to
the whole population data,

but you can take a sample.

And what you are-- what you have
in hand, again, what you have...

is a sample, and the sample has
standard deviation and variance.

And also we know how many data points
we have...that's number of sample points.

Then...

Since we chose the
chi-squared distribution for this,

we have two critical values here:
chi-squared L and chi-squared R.

And let me show you what they are.
Chi-squared is distribution related to...

...normal distribution, in fact,
the square of normal distribution.

So, it is positive and it is on the right side.

Now, chi-squared, uh...
left side is this.

And this one [on the right] is this.
Now, what are they?

So, you choose confidence level.

Remember, the confidence level was...
is a number... a small number.

Usually you choose 0.05 or 0.01,

and if it's not important,

you can choose 0.1, for example.

Most often we choose 0.05 because it's,
uh... it's better, it is the best way, I think.

Now, what is this chi-squared L?
Chi-squared L is a point on the...

...on the x-axis, such that this area here
under the curve of chi-squared is alpha over 2.

And this area on this side
[right side] is alpha over 2.

So, the area in the
middle is 1 minus alpha.

So... because you see the area
under the curve is [equal to] 1.

The whole area is 1, so if this is α/2 and
this is α/2, add them, you have alpha,

the whole thing is 1, so this area between
these two is 1 minus alpha [1 - α].

Okay, so...

Let me erase this, and...

No, let me write down the confidence
interval, say, end points, the interval itself.

In the end...

So, you calculated what?

You calculated s and s-squared.

And you have chi-squared L and
chi-squared R, and you also have n.

So, we use chi-squared with
n minus 1 degrees of freedom.

That is n minus 1 [n - 1]
degrees of freedom.

So...

For sigma squared, we have this interval,
so on the left side we have n minus 2--

No, sorry, that's n minus 1, s-squared,
divided by chi-squared (on the left that's R).

And on this side we have n minus 1,
s-squared, [over] chi-squared L.

They are different, so the
left side is R, the right side is L.

And for sigma... we have...

It's like almost the same thing, so sigma
is just the square root of this thing.

So, we have square root
of this n minus 1, s-squared

over chi-squared R.

And on this side, we have n minus 1,
s-squared, [over] chi-squared L.

Okay, so these are the things that
we will calculate, these two, that's it.

We have the confidence interval
for sigma and sigma squared.

So, let's look at this again.

We know this [n - 1], we know this [s-squared],
we calculate this [chi-squared R]--

[correction] we calculate
them [n - 1 and s-squared]--

and also this [on the right] we calculate, and
this gives me an interval for sigma squared.

And this [below] gives me
the interval for sigma.

So, after I give you an example,
I will also discuss...

I give you what this really means. So...

Let me erase this. And so...

The one that I chose, in fact,
comes from the book.

And that is...

Confidence interval estimate
of sigma for pulse rates.

Confidence interval estimate
of sigma for pulse rates.

Now, that makes sense because...

But, that probably has significance for...
like, maybe... I don't know. Maybe...

Well, health insurance companies...
so they gather a team to give them

a confidence interval for sigma
and sigma squared. So...

What they have to do is...

preferably, find pulse rates of
everybody in certain society,

if that was United Stated, then
everybody who lives in this country.

But, that's not feasible, right?

And that must be done at the same
time, say, so at this very moment.

So, just imagine how can you do that.

What the do is, they just...

take a sample of people in this country
at random and find the pulse rates.

So, they have a sample. Now,
this sample is given here and...

So, the sample consists
of some numbers here.

I will just write some of them,
but I have all of them in R.

Some of them look like this:
the first person has 76,

then the second 76,
then 86, dot, dot, dot.

Last one is like 66.

So, this is the sample, sample pulse
rate for certain number of people.

And we will find out everything in R.

So, let me share the R with you.

Where is R? Okay, I see.

Okay.

So, you can see that I named this thing
"data," we can call it "my data," whatever.

And these are the numbers:
76, 76, 86, and blah, blah, blah.

And the last one is 74.
Oh, okay.

So, this is my data, and I have
already entered, so the data...

Well, let's see, what was wrong?

Oh, I don't know, something
is terribly wrong here.

Okay, so yeah, I guess [we deleted some].
So, let's check data again.

This is my data.

First thing is n, I need n number of
sample points, so I enter length of my data.

So, let's check how many: 22.

So, degree of freedom (df)
is n minus 1, so df is 21.

What else do I need?
I need the s and s-squared.

So, s is my standard
deviation of data,

and "s_sq" (s-squared)
is variance of data.

Okay, let's check s and s_sq.

Well, I said s and s-squared,
so s-squared is that 's' squared, in fact.

So, let's see if s-squared
is that 's' squared.

S power 2...
See, they are the same.

In fact, variance is s-squared.
Or 's' is square root of variance.

So, what else do we need?

We need the chi-squared L
and chi-squared R.

You can find them like this: "qchisq."

Oh! [corrects himself]
Alpha, let's enter alpha.

Alpha is... How much was alpha?
0.05...or no [corrects typo of 4].

That's the usual alpha [0.05], so
I need "qchisq"-- Did I enter that?

No, there isn't "qchisq" [yet],
[corrects typo] "qchisq" alpha,

over 2, and the degree of freedom (df).

Let's call it something...
Let's call it... "chi_L," right?

So, it's easier when I have to do it again.

So, that is chi-squared left, I just
called it something [chi_L], and...

chi-squared right [chi-r] is qchisq.

Now, this is 1 minus alpha over 2
with the same degree of freedom.

Okay, so...

I guess I have almost
everything here.

What is the formula for s?

For s, it was square root of (sqrt) of
n minus 1, which was degree of freedom...

well, let's write it "n-1" times
s-squared "s_sq," all of that divided by chi...

...well, that was... which one am I...
"chi_L" [corrects himself], "chi_r."

R goes to the left [so] "chi-r."

Oops, what did I enter?

Oh, yeah, I have to enter "times" [*];
otherwise, it doesn't understand.

So, 7.63, let me write
this here somewhere.

So, that's for sigma,
in fact, so 7.63 [unclear].

Let me... Oh, let me see.

And the other one is the same thing
except chi squared left.

14.17. So this one is 14.17, okay?

And I also need...
This is for... This is...

7.63 and 14.17 are
the two ends for sigma.

I need for sigma squared.
That is the variance.

So I don't have to find
the square root of them,

so this was square root of this thing,
so let's just remove square root.

And that is for variance.

So that's 58.24. Let me
write it somewhere here.

58.24.

The other one would be...

...the left one. Let's
remove the square root.

And that is for variance. 200.95.

200.95 [unclear] come back.

I guess I have everything.
Let's go back to the board.

So I calculated this one, this one,
and the other one.

And that is all I needed.

So...

Let's write for sigma first.

So I found that sigma is
between 7.63 and 14.17,

with 95% confidence.

And the other one is this one:

58 and 200,
I'll just write it here.

I'm sorry. I didn't keep
the numbers up there.

So... And sigma squared is
between these two numbers:

58.24 and 200.95,
with 95% confidence.

So I found that the standard deviation
of the population with 95% confidence

is between these two numbers.

Pulse rate doesn't have any units, right?
So that-- I guess the unit is "per minute."

I think it's per minute.

So they just count the number
of pulses per minute.

So that sigma is per minute, in fact.

So that's 7.63 per minute
and 14.17 per minute.

Sigma has the same unit as
the quantity that you are studying,

but sigma squared does not.

But sigma squared has
more theoretical significance.

But sigma makes
more sense in practice.

So that is sigma for the population
and sigma squared for the population

and with 95% confidence.

We can find, say,
0.1 percent confidence.

So let me rewrite this here
and we'll find 0.1 percent.

So this is for 95
(sigma squared for 95 is this).

Let me write this here
and then find the other one

with more confidence, say.

So this is for 95%, and let's see
if we can do that for...

What difference...
What kind of calculation?

Let's see. Well, let me
share this thing again.

Alright, so with 0.01, the only
thing that changes is alpha.

The rest is the same.

Alpha this time is 0.01, okay?

So the thing that changes
is chi-L and chi-r,

so let's calculate chi-L and chi-r again.

Chi-r is q chi squared 1 minus alpha over 2.

Okay, because I have changed alpha,
so chi-r is this and chi-L, let's find it.

Okay, and chi-L is this.

The rest of the calculation is the same,
so let's find the square root of...

Everything else is the same,
so let's see: sqrt, this is chi-L,

so that's the right side.

16.03.

16.03, let me write it on the board.

And the other one is chi-r --

Let's just change this to r,
let's just go back.

And 7.06.

Let's find confidence
level interval for 257.21.

And the other one, chi-r: 49.91.

And this is 99% confidence.

Okay, so let's go back to the...

Where is it?

So these two, let's look at these two
and see what we have here.

And why... So this one, let's take sigma.

That interval is something like,
this is say 7.63 and this is 14.17.

That is for 95%.

What is this? 7.06 is right by here,
and this is like 16.03.

So that's 16.03 and 7.06.

As you see, the interval
for 95% is shorter.

So you say that
"with 99% confidence,

I think sigma lies in this interval."

The other one is 95% confidence,
and that interval is smaller.

If you want 50% confidence,

then that will become even more,
even smaller than this one.

Why is it happening?
What's the meaning?

And why more confidence
gives you larger interval?

You see, when you
need more confidence...

How shall I say that?

Well, you see, this larger interval--

so you are more confident
that they are...

This lies in a larger interval...

Well, because it has
to do with probability.

I can discuss it using probability,

so you can say that this
is, in fact, probability.

The probability that it lies
in this interval is 0.99.

So that makes the interval larger
because when you need the probability--

And remember, probability is like
area under certain curve,

theoretically or mathematically.

So that changes the interval

or let's say the area under the curve.

But intuitively, that also makes sense

because when you extend the range
with larger probability,

you can... you are more certain
with larger probability

that it really is here.
So if you say, okay,

you have hidden something in the house,
and the house has 10 rooms.

You can say that with--
just choose 9 of them

and you can say with 0.9 probability,
that object is in one of those 9 rooms.

But if you reduce that to 5 rooms,

then you can say, okay, with 50%
probability, it is in one of those rooms.

Now, which one is much better?

The 9 rooms is much better, right?

Because even intuitively,

you can say, yeah, I think
if we choose 9 rooms out of 10,

then there's a larger probability
that the object is in one of them.

But if you choose just
5 rooms out of 10,

then the probability goes down

(the probability that object which is
hidden is in one of those 5 rooms).

So you can see that
this is like the rooms.

This [the top interval] is much less
number of rooms. This is like 5 rooms,

this [bottom range] is like 9 rooms.

Maybe this example I give you,
that example makes sense.

This nice example, I just
came up with that example.

So if you want more confidence,
you get larger interval,

or let's say larger--

I should say it backwards.

Larger interval gives
you more confidence.

Okay. So that was the confidence interval
for this standard deviation and variance,

and I hope to see you next week.