- [Instructor] We're told suppose that 15% of the 1,750
students at a school have experienced extreme levels
of stress during the past month.
A high school newspaper doesn't know this figure, but they
are curious what it is, so they decide to ask a simple
random sample of 160 students if they have experienced
extreme levels of stress during the past month.
Subsequently, they find that 10% of the sample replied "yes"
to the question.
Assuming the true proportion is 15%, which they tell us
up here, they say 15% of the population of the 1,750
students actually have experienced extreme levels of stress
during the past month, so that is the true proportion,
so let me just write that, the true proportion for our
population is 0.15, what is the approximate probability
that more than 10% of the sample would report that they
experienced extreme levels of stress during the past month.
So pause this video and see if you can answer it on your
own and there are four choices here I'll scroll down
a little bit and see if you can answer this on your own.
So the way that we're going to tackle this is we're
going to think about the sampling distribution of our
sample proportions and first we're going to say well
is this sampling distribution approximately normal,
is it approximately normal and if it is, then we can use
its mean and the standard deviation and create a normal
distribution that has that same mean and standard deviation
in order to approximate the probability that they're
asking for.
So first this first part, how do we decide this?
Well the rule of thumb we have here and it is a rule
of thumb, is that if we take our sample size times our
population proportion and that is greater than or equal to
ten and our sample size times one minus our population
proportion is greater than or equal to ten, then if both
of these are true then our sampling distribution of our
sample proportions is going to be approximately normal.
So in this case the newspaper is asking 160 students,
that's the sample size, so 160, the true population
proportion is 0.15 and that needs to be greater than
or equal to ten and so let's see this is going to be 16
plus eight which is 24 and 24 is indeed greater than
or equal to ten so that checks out and then if I take
our sample size times one minus P well one minus 15
hundredths is going to be 85 hundredths and this is
definitely going to be greater than or equal to ten.
Let's see this is going to be 24 less than 160 so this
is going to be 136 which is way larger than ten so that
checks out and so the sampling distribution of our sample
proportions is approximately going to be normal.
And so what is the mean and standard deviation of our
sampling distribution?
So the mean of our sampling distribution is just going to
be our population proportion, we've seen that in other
videos, which is equal to 0.15.
And our standard deviation of our sampling distribution
of our sample proportions is going to be equal to the square
root of P times one minus P over N which is equal to
the square root of 0.15 times 0.85 all of that over
our sample size 160, so now let's get our calculator out.
So I'm going to take the square root of .15 times .85
divided by 160 and we close those parentheses
and so what is this going to give me?
So it's going to give me approximately 0.028 and I'll
go to the thousandths place here.
So this is approximately 0.028.
This is going to be approximately a normal distribution
so you could draw your classic bell curve for a normal
distribution, so something like this.
And our normal distribution is going to have a mean,
it's going to have a mean right over here of, so this
is the mean, of our sampling distribution, so this is
going to be equal to the same thing as our population
proportion 0.15 and we also know that our standard
deviation here is going to be approximately equal to
0.028 and what we want to know is what is the approximate
probability that more than 10% of the sample would
report that they experienced extreme levels of stress
during the past month, so we could say that 10%
would be right over here, I'll say 0.10 and so
the probability that in a sample of 160 you get
a proportion for that sample, a sample proportion that
is larger than 10% would be this area right over here.
So this right over here would be the probability
that your sample proportion is greater than, they say
is more than 10%, is more than 0.10 just like that.
And then to calculate it, I can get out our calculator again
so here I'm going to go to my distribution menu right
over there and then I'm going to do a normal cumulative
distribution function, so let me click Enter there.
And so what is my lower bound?
Well my lower bound is 10% 0.1, what is my upper bound?
Well we'll just make this one 'cause that is the highest
proportion you could have for a sampling distribution
of sample proportions.
Now what is our mean well we already know that's 0.15.
What is the standard deviation of our sampling distribution?
Well it's approximately 0.028 and then I can click Enter
and if you're taking an AP exam you actually should write
this you should say, you should tell the graders
what you're actually typing in in your normal CDF
function, but if we click Enter right over here,
and then Enter, there we have it, it's approximately 96%.
So this is approximately 0.96 and then out of our choices
it would be this one right over here.
If you were taking this on the AP exam you would say that
called, called normal normal CDF where you have your
lower bound, lower bound, and you would put in your 0.10
you would say they use an upper bound, upper bound
of one, you would say that you gave a mean of 0.15
and then you gave a standard deviation of 0.028,
just so people know that you knew what you were doing.
But hopefully this is helpful.