BAM! Mr. Tarrou Here to finish our discussion of binomial setting and binomial distributions.
Since a binomial setting or binomial distribution is a class or type of discrete random variable
you can set up pdf tables using the binompdf command in your calculator. It will be under
the Distribution menu. So we have got the binomial distribution of B(n,p) or n is 5...
there is going to be a total of 5 number of trials, and a 30 percent chance of success.
And we are defining x as the number of success. So the value of x can take on zero successes
out of five all the way up to five. You can have a success for every trial. And our pdf
table has got probabilities of 17 percent, 36 percent, and so on. All I did was just
type in 0, 1, 2, 3, 4, 5 into the binompdf command in my calculator to get all of those
exact probabilities. Now you can also accumulate those probabilities and make a cumulative
function. This type of cumulative probabilities are what allowed us to make Ogive graphs when
we were learning about histograms. I got those by doing binomcdf with the n,p,k. k is the
number of success. You could just add all of those up. I rounded these off pretty badly
so my last two cells show a probability of 1. That is just round off error. So just be
aware of that. Now let's talk about how we can handle binomial counts, or binomial distributions
without this idea of it being a discrete random variable. Actually it is, but if you have
a large enough trial... or if you have enough trial that is too large to maybe use your
calculator with and use these binomin pdf and cdf commands you can do something called
Normal approximation. Also our graphing calculators can only work binomial distributions when
they are in terms of counts. Later on we are going to learn about how to deal with binomial
proportions. That is the exactly the same setting, only x is going to be defined as
percentages of success and not number of successes. Since our calculator cannot work with binomial
settings in terms of proportions, this idea of Normal approximation... which actually
is not written on the board yet... but it will be on the next screen... is pretty important.
Ok. So how do we find the mean and standard deviation of a binomial count? The mean of
this binomial count is n times p. And the standard deviation is the square root of n
times p times 1-p. If you are taking an AP stats class and you are going to take that
test at the end of the year to try and get some college credit, these two formulas are
on that formula sheet when you take the exam. Now if you are talking about mean and standard
deviations, especially standard deviation is not a valid measurement unless your distribution
is Normal. Remember we talked about Normal probability plots. That was a graph that our
graphing calculators could make automatically, or your computer, to help check that your
distribution... your univariate quantitative distribution is indeed Normal. So look that
up. Normal probability plot of you have forgotten. We have already covered it this year. Well,
the Normality check for the Binomial setting is right here. If n times p is greater than
10 and n times 1-p is greater than 10, then your sample is large enough for your binomial
distribution to be... well... approximately Normal. And thus you can let the bell curve
and the z score calculations approximate your answer instead of getting an exact answers
you would get from binompdf and binomcdf. Let's see what that Normal Approximation looks
like. WHOOO! [nanananana] There we go. Ok. Normal Approximation uses z scores
to estimate probabilities of desired events like the probability that x, again the number
of successes is greater than a number... less than a number. Whatever. So our example we
are going to walk through now: During the years biggest holiday, I did make up these
numbers by the way, it is estimated that 68% of people travel to see family. So if you
survey 350 people, what is the probability that over 250 people in your survey say yes
they do travel to see family? Well, step one is to make sure that it is even a binomial
setting. Just because it is a yes/no answer does not mean, you know... there are two outcomes
for each trial, does not mean that you are automatically mean that you are a binomial
setting. You have four checks for that. Make sure that there are only two outcomes. Make
sure that you have a set number of observations. We are going to ask 350 people. We are not
going to keep going until we get our first yes. We are not going to keep going until
200 people say yes. We are going ask a set number of people, 350, and so n is fixed.
Independence. Well, usually your statement there is, I am doing an SRS and the population
is over ten times the sample size. So we have independence so we have independence in this
setting. My probability of success is constant. It is going to be assumed to be a constant
68%. And this fifth check is not for the binomial setting, but this fifth check is for Normality.
And again, we cannot do Normal approximation unless our distribution is actually Normal.
So that is n times p is greater than or equal to 10, and n times 1-p is also greater than
or equal to 10. Both of those checks have to pass. Otherwise your distribution is not
Normal and you will be forced to use binompdf or binomincdf in your calculator. That is
not really such a bad option, as long as you are within or working with binomial counts.
Binomial proportions cannot be handled by... you might be using computers and Minitab.
I am not really sure how those work, but I know your graphing calculators will not handle
binomial proportions. Alright. Let's go onto the next screen and see what this example
looks like! Whooo! [nananana] Ok. I humor myself! Top three lines are in green because
these top three lines are not Normal approximations. No, they are coming from our calculator and
let's just see what the answers come out to. The binomial... or the probability that x
is greater than 250 is found by doing 1 minus binomcdf... if you have an INSPIRE, it will
count up. But if you have a TI-83 or a TI-84 to help you with this class, they DO NOT count
up. I think I said something wrong. If you have a TI-NSPIRE it will do these inequalities
whether they are less than or greater than, and you don't need to use this subtraction
idea that I have here. But most people use TI-83 and TI-84, and for binomial count distributions
they are only capable of counting down. So I am having to do 1 minus the binomcdf, the
compliment of x being greater than 250... the compliment or the opposite of having 251
or 252 or 253 and so on successes, is to have 250 or less. So this command is counting down
which is what our TI-83 and TI-84 are capable of doing. Those are the calculators that most
of my students use. So it is 1 minus binomcdf(350,.68,250) and it comes out to be 7.49 percent basically.
Now if I change that greater than symbol to greater than or equal to, and this... we are
handling this as a discrete random variable... and that is what our calculators with the
exact value commands will do, it will change the answer slightly. Because there is a probability
of x equalling exactly 250. So 1 minus binomcdf... the opposite... you know with n, p, and the
number of successes. The compliment of 250 or more, the compliment or the opposite of
250, 251, 252 successes and so on is 249 or less. So my last number has changed here a
little bit and my final answer has changed. 9.3 percent approximately. What would that
look like as Normal Approximation, using a z score calculation. As soon as you set up
a bell curve and you calculate the mean and standard deviation of that variable, now you
are using that bell curve... a mathematically model... to approximate the actual answer.
And this is a smooth continuous random variable. Did you hear what I just said? The bell curve
represents a smooth continuous random variable. It is not going to be handled as a discrete
random variable like a binomial pdf or cdf are calculated with your calculator. So it
is Normal Approximation even though the binomial setting is technically is a discrete random
variable. Once you go into Normal approximation you are using z scores and bell curves, this
now acts as... at least as the approximation goes... a continuous random variable. So the
probability that x is greater than 250 is equal to... The z score which is estimate
minus parameter over standard deviation, that gives me a score of 1.37. Don't forget the
unit of measure for z score is the number of standard deviations your statistic is away
from the mean. So 250 is 1.37 standard deviations to the right of 238. If you are using a chart
to get your p-values, I stress the use of those. I feel like they build the basic understanding
statistics better than just using a calculator. So I am getting area shaded to the left in
the... the chart that my book uses. So the entire area in the density curve. I am using
some... we are pulling back some old concepts from chapters 2 and 3 of my book, or you know..
topics I have discussed in videos quite a while back. So 1 minus .9147, this upper tail
area therfore has an area of .0853. And our summary statement is, "The probability that
more than 250 people say that they travel to see family out of a sample of 350, so I
am not really thrilled with my summary now that I am reading it.... The probability that
more than 250 people say yes they travel to see family out of a sample of 350 is approximately
8.5 percent. Now just to reiterate one more time that is a Normal approximation and we
are using a smooth continuous bell curve to make that Normal approximation... if I were
to change the wording and say that I want the probability that x is greater than or
equal to 250, that would not change my final p-value. That would not change my final probability.
It will however require me to change my wording. Your wording, your language has to match your
notation. So now it will be the probability that 250 or more people say that they travel
to see family out of a sample size of 350. So there you go. We finished our discussion
of binomial distributions. Now go do your homework. WHOOO! BAM! Thank you for watching.
I do appreciate having the opportunity to help you out.