Welcome to Excel 2013, Statistical
Analysis video number 45.
If you want to download this workbook, or the PDF files
and follow along, click on the link below the video.
Hey, chapter seven, our last topic,
is sample proportion and the sampling distribution of p bar.
Now, sample proportion, we've already seen a lot of these,
is just p bar, when I take x divided by n,
where x is the number of elements
in the sample that possess the characteristic of interest.
We might ask, do you balance your checkbook, yes or no?
Did you sit in traffic, yes or no?
Did you respond to the letter asking
for donations, yes or no?
It sounds like a binomial experiment.
And it is.
And n, of course, is the sample size.
So we're going to go out and take a sample
and get some proportion or percentage.
The sampling distribution of p bar,
if we went through the same steps as we did last video,
we could create it.
And we would discover the same things.
It's normally distributed.
There's less variation.
And p bar equals exactly the population p.
Now, there's one thing we have to check.
For the sampling distribution of p bar
to be approximated by the normal distribution,
we have to take sample size times our proportion.
That has to be greater than or equal to 5.
And we have to take that same sample size and 1
minus the proportion.
And that has to be greater than or equal to 5.
If both of those are true, then we
can use the normal distribution.
As we mentioned just a second ago,
expected p bar equals population proportion.
Now, the standard error of the proportion or standard
deviation of p bar, here it is right here--
p divided by the component of p divided
by n and the square root.
Oh, and there's that correction factor.
And the same rule applies.
If sample size divided by population size
is less than or equal to 0.05, then we
do not have to use this correction factor.
We can just use this standard deviation sub p bar.
Now, let's go over to Excel.
Our first example, we're going to assume the population
proportion p is the proportion of Americans
that balance their checkbook.
And that's equal to 0.56.
If the population in the USA is 300 million,
and we take a sample of 400 Americans,
answer the following questions.
The questions are down here.
What is the probability that the sample proportion
will be within plus or minus 0.02 of the population
proportion?
And what's the probability that it will
be within plus or minus 0.04?
Exactly like last video, except for now we're
using the central limit theorem and proportions.
Now, here's our p, so our 1 minus p equals 1 minus p,
not balanced checkbook.
If this is the population parameter
that Americans balance their checkbook,
then 0.44 do not balance their checkbook.
Hey, there's our population n, our sample size.
We can go and check and see if we
have to use our correction factor equals little n divided
by big N.
And it's much, it's teeny, teeny.
We could have eyed that one.
We didn't even need to calculate it, much too small.
Now we can do our standard error.
Equals the square root.
And we have to do p times 1 minus p.
And we already calculated that.
So those, and then divide by our n.
Now, the multiplying in here will
work left to right and then division.
So we simply close are parentheses
and enter that as our standard error.
For the sampling distribution of p bar,
that is going to be our standard deviation.
And the original population parameter proportion,
that's going to be our expected p bar.
Now, can we use normal?
We have to have these two checks.
So equals n 400 times our p.
Is that going to be greater than 5?
That's going to be gigantic.
Equals the complement of p times our n.
Is that going to be bigger than 5?
It's going to be gigantic.
That means we can use our norm.dist with our proportions.
Now, what is the expected p bar?
Oh, yeah, it's just given.
Enter.
Now, we can calculate our two probabilities.
Margin of error, that's the amount
on either side of our expected p bar that we add.
So we're going to get 0.54 and 0.58 equals our expected p
bar minus the margin error.
Tab.
Equals our expected p bar plus our margin of error.
Tab.
Now, we have a p bar upper and a p bar lower.
So we can use norm.dist.
And oh, look at this.
The screen tip says x.
Well, we've used norm.dist with xs, with x bars,
and now, we're going to use it with p bar.
So no problem.
The bigger p bar, mean.
It's not a mean for an x or an x bar.
It's our expected p bar.
Comma, and then our standard deviation.
Remember, this is the standard error.
This is a standard error of the sampling distribution of p bar,
cumulative 1.
That'll give us everything in our curve up to 0.58,
but we need to subtract another norm.dist.
Oh, there's the p bar we put into x, the mean.
That's our expected p bar, our standard deviation.
That's the standard error of the sampling distribution of p bar,
and then one close parentheses.
And Enter.
And there's our probability.
The probability that the sample proportion, our p bar,
will be within plus or minus 0.02 of the population
proportion is about 58%.
Now we could do it for the other one.
Equals-- well, we got to go get our p bar minus the bigger
margin of error tab equals.
We've got to go our p bar plus the bigger margin of error.
Tab.
And now, are two norm.dists.
We're going to take the upper p bar, our expected p
bar, our standard error, 1 close parentheses, and the norm.dist.
Hey, not x, we're going to take our lower p bar, not
the mean, but our expected p bar, and our standard error,
1 close parentheses, and Enter.
The probability that the sample proportion or p
bar will be within plus or minus 0.04 of the population
proportion is about 90%.
So if we went out and took a sample, we're about 90%
sure that our p bar would fall between 52% to 60% of Americans
balance their checkbook.
Hey, we want to go look at example two, p bar 2.
Now, in this example, the p of the population proportion
is an insurance company's claim that they settle claims
within one week at a rate of 75%.
So if the number of policies issued is 35,000, we go out
and take a sample of n equals 300.
We want to calculate the probability
that our p bar would fall within 0.04 margin of error
of our population proportion.
Now, we don't really need to calculate this,
but we can do this because 300 divided by 35,000
is going to be much smaller than 0.05.
That just means we don't have to use our correction factor.
Our expected p bar, hey, we were given that.
And Enter.
The standard error for the sampling distribution of p bar,
hey, we'll take the square root and better get that p.
And I didn't calculate 1 minus p,
but I can simply do it right here in the formula,
times 1 minus p.
And divide it by our n of 300.
So that will be Enter.
Our standard error of 0.027.
Now, we can calculate our p bar 1 and 2 with our margin
of error equals, hey, there's our expected p bar minus 0.04.
Enter.
Equals expected p bar plus our margin of error.
Now, we need to test whether we can
use the normal distribution.
So I'm going to test.
And I can already see it's probably going
to be pretty big numbers here.
So we got our p 0.75 times our n, Enter, our 225,
and the complement 1 minus p.
I did Control up arrow there.
Close parentheses times our n.
And we get another big number.
We can use the normal to approximate the binomial.
Now, we want to the probability between these two proportions
for finding a p bar.
So equals 0, and it's norm.dist again.
Our upper p bar, 0.79, our expected p bar,
our standard error, 1 close parentheses.
And then we subtract another norm.dist.
That x?
Oh, yeah, that's the lower p bar,
our mean is going to be our expected p bar.
Standard deviation is our standard error,
1 close parentheses and enter.
So the probability that we could go out and get a p bar that
lies between 0.71 and 0.79 proportion of claims settled
within week is about 90%.
All right.
In this video, we talked about p bar and the sampling
distribution of p bar.
And we talked about how to use their normal distribution
to approximate and make probabilistic statements
about finding a p bar within a lower and upper value.
We saw this insurance example here.
And then back on the p bar, we saw this balancing
your checkbook example.
All right.
That's it for chapter seven.
Next chapter, chapter eight, we'll
talk about confidence intervals.
All right.
We'll see you next video.