BAM!!! Mr. Tarrou. Alright. We are going to start our example of working with binomial
proportions. And just as a footnote, sorry about the little correction I had to make
on the last video. I wrote the mean of p is equal to p hat. I had those backwards. Ok.
We are going to use the same example that I used in my video about binomial counts for
the binomial setting just so you can see the close relationship between using binomial
proportions and binomial counts. There are really exactly the same. I can't really say
that because the formulas are a touch different. The mean and the standard deviation formulas.
But other than, it is still a binomial setting. You saw the same checks... the same checks
for normality and we are going to be doing z scores again for Normal Approximation. So
during the year's biggest holiday it is estimated that 68 percent of people travel to see their
family. I just made these numbers up by the way. If you survey 350 adults, or just people
in general, what is the probability that... now I am doing the 250 over 350 because I
am just showing you that I am just taking the count value that I had in the original
example and I am dividing it by 350 to convert it into percents. So if you survey 350 people,
what is the probability that 71.4 percent say that they travel, or 71.4 percent or .714....
the proportion of people.. say that they travel to see family. Should probably add that in
there. Well, if you are going to do a Normal Approximation you first have to make sure
that it is a binomial setting. You have to make sure that it is actually approximately
Normal. Which now that I am talking I forgot to put that in here. So I will add that in.
Of course you need a mean and a standard deviation. So the mean of p hat, the mean of our sampling
distribution should be the same as the population's means.... as long as the sample is unbiased.
Certainly we do want it to be biased. The standard deviation of p hat is the square
root of p times 1-p over n. And while our sample proportion, or in the question it is
about 71.4% of the people saying yes, we are told that indeed 68% of the people travel
to see their family. So it is going to be the 68 percent that we use in our z score
calculation. Remember you do not use p hat ever in this standard deviation formula unless
you are doing specifically a confidence interval. That is not what we are doing right here.
So I have my four checks for a binomial setting. We have two outcomes of traveling or not.
We have a set number of observations of 350. We need to have independence. Our statement
there is generally going to be the SRS is assumed because I did not actually say that
I took an SRS of 350 people. And the population must always be greater than ten times the
sample size. That is what you need for independence especially. We also need to have a constant
probability of success. And we do if we are polling from a large population. And you know,
that actually is truly 68%... we have a constant probability of 68 percent. Now if I am wanting
to do Normal approximation, then all calculations with binomial proportions do involve z score
calculations which we call Normal approximation. Well if you are going to do that you need
a Normality check. You need to make sure that 350 times the probability of success is greater
than or equal to 10, and that n times 1 minus the probability of success is both greater
than... both of those have to pass.. they both have to be greater than or equal to 10.
And they are. I have checked that I have a binomial setting, I have checked for Normality...
I have not written it because I am a little bit out of space here. But this is my Normality
check. And let's walk through some examples. I am also going to show you a tie in between
working with binomial counts, those binomial cdf and binomial pdf commands, as well of
course hit the main purpose of this video. This example of working with binomial proportions.
Again how do you a z score calculation with this type of question. Alright!!! [nanananana]
Alrighty. So we are going to ask the question, or rephrase it from the previous screen. What
is the probability of our sampling proportion exceeds .714? Yes, so I have drawn my bell
curve. You always want to do that when you do z score calculations just to make sure
you let the person grading your test.. you understand what the population proportion
is... where that goes. The population mean. Your sample statistic, where that goes. Which
direction you are going to shade for the wording the question. And again, showing your work.
I am using tables, the hand calculations of z scores and the table. So we have your standardized
test statistic, that is what it reads on your formula sheet, or my z score is equal to estimate
minus parameter over the standard deviation of the estimate. We have already calculated
that on the previous screen. We get a z score of 1.36. That should sound realistic or correct.
Well, my statistic is to the right of the mean. When we are on the right of the mean
the z scores are positive. Let's just remind ourselves the unit of measure for z scores
is the number of standard deviations from the mean. So .714 is 1.3 standard deviations,
or 1.36 standard deviations to the right of the mean. At any rate. That gives a p value
of .087, or there is 8.7 percent probability of having more than 71.4 percent of our sample
say yes they do travel to see family during the year's biggest holiday. Let's do an inequality
of an interval question. We want to find the interval inside of a bell curve. I have not
done one of these for a while in a video. What is the probability that between 66 and
71.4% of the sample says yes they do travel to see family. Well when you want the area
of an interval in a bell curve, it is the area to the left of the large number... So
the area to the left of .714... minus the area to the left of the small number... so
I don't want this left tail area shades. I am going to find that left areas off the table
and subtract them. So the area to the left of the large minus the area to the left of
the small value. So I picked the same number. The z score again is 1.36 and the area to
the left of that value of .714 is approximately .913. And then we are going to do a z score
calculation again for the small number, or the number on the left. So z equals estimate
minus parameter divided by standard deviation of the estimate. I am using the words from
your AP formula sheet just to get you used to that notation, or that language again.
We have a negative z score because we to the left of the mean. We are less than one standard
deviation away from the mean, negative .8. That left tail area that we pull out of our
chart is .212. So from this .212 is the area to the left, this tail area that is not shaded
because we are going to subtract it out. And when we do that subtraction of .913 minus
.212, it comes up to .701. I did not write the summary statement because I wanted to
get as many examples done in this one video as I can. There is approximately 70.1 percent
probability between 66 and 71.4 percent of my sample say "Yes, they travel to see family."
And again, any time you do a z score calculation, or you use Normal Approximation... or you
use bell curves, that is a continuous random variable even though the binomial setting
is a discrete random variable. We are counting the number of people out of the 350 that say
yes. We have converted that into a decimal, but remember that when we studied binomial
setting and binomial counts the first time that was a form a discrete random variable.
Any time you use a z score model, that is a mathematical model... perfect bell curve...
and that puts us into the realm of a continuous random variable. So just for argument sakes,
if I change this and I add a couple of equal signs into my inequalities that would not
change my answer from the Normal approximation. So let's compare that fact where with this
continuous model when I change the inequalities it does not change my numerical. It does need
to match the wording of the question. How does that vary with what I have got down here
in blue? Now, what I am doing down here in blue is I am showing you a couple of things.
I am reminding you about the binomcdf command that you learned when we were learning about
the binomial setting and binomial counts. I am also trying to try in... you know...
this example from the previous video. And I remember from the previous video I never
did any examples about finding the interval between two numbers. So I have got the probability...
First of all, I have done 66 percent times the sample size which is a count equal to
231. Changing x from the percent of people that say yes, to the number of people that
say yes. So 66 percent... 66 percent of our sample size of 350 is 231 people. 71.4 percent
times 350 is almost, it was like 249.9 or something like that, so that is 250 people
saying yes they travel to see family during the biggest holiday of the year. So what is
the probability of x, because now I have converted it to a count to tie in with the binomial
settings with counts and proportions again... What is the probability x is between 231 and
250 without the inequality signs. Well, binomcdf... that is under 2nd Distribution in your calculator...
binomcdf(350,.68,249). Yes, because I want x to be less than 250 and the first count
value less than 250 is 249. Now minus binomcdf(350,.68,231) because I want to cut out... The count values
that I want to be basically 232, that is greater than 231... I want them to be 232 up to 249.
And it comes out to be 68 percent. That is a little bit lower than we got from the Normal
approximation. Now if I add in the equal signs and I want x to be in-between 231 and 250
but including those numbers, I added the equal sign, now my binomcdf statements from my calculator...
I want 250 or lower, so still finding the area to the left of the larger number. But
I am using 250 now because of the equal sign. And I want x to be greater than or equal to
231. So I want 231 to be included in the answer. So 231 or greater cuts off 230 or less. So
that is my last k value in my last binomcdf statement. And when I subtract those in my
calculator I get .731. So you see how if I don't include the equal signs I am a little
below what I got from Normal approximation. If I do include the equal signs for this particular
example, I am a little bit above. So that is how that Normal approximation works. It
does give you a pretty close answer, but it is not exact. That is why it is called Normal
approximation. And changing the inequalities with a continuous random variable, it is important
to follow the language of the question, but it does not effect the final probability.
Alright! We have one more screen, or page of examples. Here we go! Here we go... Here
we go. [nanananana] Bam! Alright. What are the effects of increasing the sample size?
So I am going to back to my original example of what is the probability the sample proportion
exceeds .714 ? That is the same probability as before when we had an 8.7% chance of observing
a statistic larger than .714. But, now I am going to make my sample size 1400. It is a
much larger sample size. Let's do a little bit of algebra review. This sample is four
times larger than the original 350. So the standard deviation should be reduced by half.
Here I have taken... or added in a 4 where I am multiplying the sample size by four.
I am just using the standard deviation formula for binomial proportions. And we can take
the square root of 4. So pulling that 4 and square rooting it. It is in the denominator
where n is. The square root of 4 becomes 2. So here is my original standard deviation
formula with the 1/2 out front. So I am going to increase the sample size a multiple of
4, but that ultimately is going to decrease the standard deviation by half. So I am recalculating
the z score calculation with of course a different sample size. I get a smaller standard deviation
which puts me... you know.. it gives me a z score now of 2.72 standard deviations instead
of the 1.36 that we had before. So it is still the same observed difference, but with the
larger sample giving me a smaller standard deviation I am actually more standard deviations
away from the mean. That is going to decrease the area in the tail. And now I can say that
the probability of more than 71.4 percent of my sample saying yes they travel to see
family during the holidays, assuming that the original 68% was true, is approximately
.33 percent. That is a very very probability. That means that if I were to observe that
71.4% or higher, then that is very strong evidence that the given information was false.
You always set up z scores and significance test, that is a little language... I am getting
a little bit ahead of my self with the language, you always do your significance test assuming
the given information is true. Well if it was true and I did observe a sample proportion
of .714 and that did give me supposedly a very very small probability of occurring...
It was a very small probability of occurring if the given information was true. So this
is what you would call a statistically significant event. The larger sample size reduces standard
deviation making the observed difference very less... much less likely... therefore a more
statistically significant event. We are going to be talking about this concept a lot more
in the following chapters. It is the statistically significant events that allow you to question
the truthfulness or the validity of the given information. If it is said that something
is true, and you observe an outcome that was very very unlikely... it could be by chance.
People do win the lottery. But, with statistics we have to go with the odds. If we observe
an event that was very very unlikely, we must conclude that there is evidence... or it is
showing that there is evidence that the given information is false. If we assume that truly
68% of the people travel... we observe that 71% and assuming that the 68% was true giving
us this very very tiny probability would have us question... Is that 68% you told me that
travelled, is that accurate? Or maybe is it a little bit higher. Alright! So that is it
for now. We will talk more about this concept in the following chapters. I am Mr. Tarrou.
BAM! Go Do Your Homework:)