In Supplement D,
on slide 6 and 7,
we're taking a look
at a required number of observations. If we really want to have a good
understanding, how many observations do we really need to take?
The formula for this,
is "n" is equal to Zs
divided by Ax,
that full quantity squared.
Well, what do these numbers mean?
That z is a z score.
Do you remember way back when, when we took a look at confidence levels,
etc., you should have done this in a stats class, you should have done it in
um...
ECON 2070 or 2080,
there's a variety of places where you're looking at z scores. That's what this is.
Confidence levels and z scores.
For this class, we'll be looking at confidence levels, confidence levels of
ninety five percent, which equate to a z score of one point nine six (1.96) and on your
formula sheet, if you would like to use that please put that down.
"s" is standard deviation.
We've looked at standard deviation, we've learned how to compute standard deviation,
because in reality,
it will...
won't be given to you, you will have to find it yourself.
A is accuracy.
What percentage of accuracy do you want this is, within ten percent, within five
percent, within fifteen percent,
how accurate do you want this
formula to be?
And 'x' is the mean
that we're looking at.
So, we are taking a look
at finding 'n.'
A job has been time studied for 20 observations. OK, so we've already taken
twenty observation and a mean actual time
was five point eight three (5.83) minutes
and the standard deviation
of the times is estimated to be two point zero four (2.04) minutes. This when
they're being nice enough to give it to you.
How many total observations should be taken for a ninety five percent
confidence
that the mean actual time has been determined within ten percent?
So, looking at this
let's take a look
at our formula
and I think I wrote that down wrong on the slide cause it should be x-bar, not 'n.'
So, 'n' is equal to
z times s over A times x-bar
squared,
our mean.
OK?
We know
that
Z is the ninety five percent confidence level is going to be one
point nine six (1.96).
It says
that our standard deviation
is at two point zero four (2.04).
It told you that
the mean or the average
of what's already been taken
is five point eight
three (5.83) and
that should be
x-bar, and we're taking a look at accuracy here. We said within ten percent, which equates
in a formula two point one zero (0.10)
I call this plug and play.
We're gonna take the numbers we have here, and plug them into our formula.
'n' is equal to
one point nine six (1.96)
times two point zero four (2.04)
divided by
point one zero (0.10)
times
five-point eight
three (5.83),
and I want that whole quantity squared.
So, 'n' is going to be equal to, if I do the math, one point nine six (1.96) times
two point
zero four (2.04)
I should get
three point nine nine eight
fou (3.9984)
and divide that by point one zero (0.10) times five point eight three (5.83) which is point
five eight
three (0.583). Again, all of this will
be squared, so if I take what's inside the parentheses and divide
three point nine nine eight four (3.9984) divided by point five eighty three (0.583), divide my
numerator by my denominator,
I should get six point eight
five eight three (6.8583)
I want to square that number, which will be forty seven
point zero three six two
uh... eight (47.03628).
Now, can you take a partial observation?
No, so the answer here is not forty seven observations because normally we'd round
down.
When we're looking at the number of observations that need to be taken
in order to get a confidence level
of one point nine six (1.96), our z, ninety five percent confidence level, or a z of
one point nine six (1.96)
and be within
ten percent accuracy, we need to take a little bit more than forty seven
observations. Well 'n' can never be a partial observation, cause you can't take a
partial observation.
So, 'n' has to be
forty eight
observations (48).
We've already taken twenty (20), so we need twenty eight (28) more observations. We can
count the ones we've already taken to be able to get this information and get
the mean
and get the standard deviation, etc.,
but we have to be able to
take a full number, a whole number
and not a partial,
not a decimal, or a fraction, of an observation so in order to get our
confidence level at ninety five percent, or a 'z' score one point nine six (1.96), in order
to be within ten percent (10%) accuracy,
we need to have
forty-eight (48) total observations or twenty eight (28) more.
That ends supplement D's
problem on finding the number
of
observations you need to take.