Instructor: Hello, students.
This is our second example for Chapter 6,
and in this example we are going to go over
how to calculate confidence interval
for a single sample of dichotomous data.
The example here states that we have 795 males
and we have asked them if they have ever been told by a doctor
that they have symptoms of prediabetes.
The results we see here is
that 374 have told us that,
yes, they have been told
and 421 responded negatively stating,
they have not been told.
What we want to do here is
find out the true proportion of male
who have been told that they have
symptoms of pre-diabetes
based on the population represented by the sample.
So now the formula for this will come, of course,
from our textbook, and that is from page 118.
And I'm first gonna write the formula here.
So here, we are using the Z distribution
and the 95% is what was given in our question.
So before we can start solving this equation,
first, we need to find out what is P hat.
So that's what we are going to find out here.
P hat is,
again, basically 374.
That is the number of males who responded positively.
And the denominator here is the
addition of both 374 and 421,
which is our sample size.
So when we crunch the numbers here,
what we get is 374 divided by,
again, I prefer you do these calculations
with me as I'm doing it.
So let's add 374 and 421.
We are supposed to get 795.
And yes, that is exactly what we got.
And then, when we do the division here,
we get 0.47.
So now what we need to find out is the Z value.
So we go to the back of our book, Table 1B,
and the critical Z` value here is 1.96.
So now let's insert these numbers in our formula.
P hat is exactly what we just calculated.
The Z value here is 1.96,
which we got from the back of our book.
And then we have to calculate this part, which is 0.47,
then 1 minus 0.47
divided by the n, which is 795.
So let's see what we get here as we crunch the numbers.
So the numerator here will be
2.49.
So it's 0.249.
And the denominator here is still 795.
So let's see what happens here.
Now, 0.249
divided by 795.
We get, wow, a very small number, 0.0003.
So now we have to do the square root of that.
So what we are going to get here is,
I'm going to write it down.
Again, this is still the same 0.47,
and here is going to be 1.96 multiplied by,
so if we multiply 1.96,
we get,
again, I'm writing it down,
0.033.
And I have already posted an announcement regarding
decimal places and how far we want to go.
So based on that, this is how far we are going.
So now, when we do the subtraction
as well as the addition here, this is what we get.
Oops.
0.47
minus 0.033 is 0.437,
0.503.
So basically, what we can state is
that our calculation tells us that
the full population of men represented in the sample,
the proportion of those who have been told
that they have symptoms of prediabetes
with 95% confidence is between 0.437 and 0.503
or 43.7% to 50.3%.
So basically, I converted the decimal to a percentage.
So that's all for this problem.