Bam! I’m Mr. Tarrou. Back for part 2. Comparing and really defining the differences between
the two types of variables that we are going to learn, or the two types of variables we
have in statistics, continuous random variables, which granted is most of the type of problems
that we deal with and a discreet random variable. And we have this pdf table set up for the
number of TVs in a household. So, we are about to scan over the next panel of notes, and
we are going to be talking about continuous random variables, that means we are going
to bring up one more time bell curves, and z-score calculations, and that z –score
calculation is going to get repetitive, or get repeated many, many times now in the second
half of our intro. to statistics class, but let’s remind ourselves of finding, maybe
some probabilities that we did in our probability chapter, the last chapter. Like what is the
probability that x equals let’s say, one. What is the probability that a household has
exactly one TV in the house? Well, that’s 16% based on this data, which honestly I did
make up for this lecture. What is the probability that x is greater than 1? Well, that’s going
to be, well, all my outcomes that are greater than one, just like we did when we first learned
probabilities and we set up these probability models, and the last chapter more than 1 is
just .37, not it’s .57, 58, 68, 75, 76%. So, the probability, based on this data, that
a household has more than one TV is 76%. What is the probability that a household has one
or more TVs? You know, I bring in this equal sign, and now I ask for the probability. Well,
that’s going to be the compliment, the probability of one or more is the compliment of having
no TVs. And having no TVs, well that’s 8% of the population, we are guessing or estimating
form our sample, so the probability of having one or more is 92%. Okay. The reason why I
am bringing up these probability questions and reminding you of some of the questions
you might have learned when you learned basic probabilities, as basic probability rules
last chapter, is we’re going to now move into the topic of a continuous random variable.
And a continuous random variable is when we did the z-score calculations, and when we
talked about z-score calculations we have to be reminded of how to prove and that data
in normal, and don’t forget we have univariate quantitative data that Is normal, that’s
mathematical model. And when we have this perfectly mathematical model that can take
on an infinite number of x-values, you know, you would not actually be able to find something
like the probability that x equals a specific number. You would not be able to find these
two probabilities if it was a continuous random variable or an estimation used or made by
musing the z-score calculation, these would actually be the same answer in a bell curved
z-score calculation, but when you have a nice pdf where you can just pull probabilities
out of, yes indeed you can find the probability that x equals a specific number, and these
two inequalities, x is greater than 1 and x is greater than or equal to 1, of course
you’ll get different answers because we can see that probability that x equal 1 does
have value and thus changes the answers we introduced this equal sign. Okay? So, enough
of this panel of notes, let’s go to the last one. Na-an-na-na. Okay. So, bam! Last
page. It’s very easy to make histograms by hand when given a probability distribution.
Just that we just saw with the numbers of TVs in the single household. Because they’re
telling you, here’s your x values, 1, 2 , 3 very precise, and hey by the way the probability
of the zero, or the probability of a 1 or the probability of a 2 is blah-blah-blah.
So, when you are given such nice, clear information, if asked to make a histogram, please just
make it by hand. Don’t bother making the calculator, or having the calculator do it
for you. With discrete random variables, the x values being nice and clear and concise
individual outcomes, it also makes a little more sense to just out those number right
in the center of the histograms. Normally, our numbers are on the divider lines, so if
you have forgotten some of your concepts from very early in the chapter like how to set
up a histogram, or how to calculate z-score values, you might want to go back and re-watch
those videos as well, to remind yourself how to do these problems, or of course, read you
book, hopefully that will help as well. Okay. Type two variable. Continuous random variable.
Takes on all values in an interval of numbers, all values not specific countable values,
so within whatever interval is appropriate for your word problem, maybe it’s just simply
10 to 20, the x variable, or the random variable, can take on all and any values between you
know, whatever you’re min and max value, if there even is a hard set interval, which
in a continuous random variable by theory, there really is not. So, there is an infinite
number of possibilities that x can take on, or values. The probability distribution cannot
be put into a nice, clear pdf table like the discrete random variable. So, that means that
we can only describe a continuous random variable with a density curve. And you remember what
density curve is right? It does have to be a nice bell shaped curve, a density curve
could be in the shape of a rectangle. It could be in the shape of a triangle, and if it is
a shape of a rectangle, then every value between let’s say 0 and 2 is an example, would have
exactly the same possibility of occurring because the graph is flat along the top. All
you need for a density curve is that the area equals one. So, if you have a rectangle with
width of 2, its height is only going to be ½ because the area must be one. That’s
really the only defining characteristic of a density curve, the area is 1. Now, if you
have these crazy geometric shapes, I don’t know if a rectangle is all that crazy, you
just find the area by finding length times width. I just said find the area by doing
length times width. If you have a bell shaped density curve, you have to find the area under
the curve by doing the z-score, and then looking up that z-score formula to find, maybe find
some tail areas. If you have just a rectangular density curve that represents a completely
random variable where every outcome is equally likely, you just do length times width. Or
maybe you have a triangular shape, and you’re doing ½ base times height. So, don’t think
every time you see the word density curve, you do a z-score calculation. That is only
when you go back and you start reviewing the idea of a density curve that is normal. Okay?
So, if you have a normal density curve, then your z-score is equal to your statistic minus
your parameter over the standard deviation of your statistic. You will, we’ve really
only seen this equation in the form of x minus mu over standard deviation, but now we’re
starting to get into the textbook, and really getting close to the real meaty questions,
where we’re going to work with means, and two means at the same time, and proportions,
and hey, by the way do you remember that previous window where I showed you how to find the
mean and standard deviation of the discrete random variable? All of those little like
miniature formulas, and all of those formulas for standard deviation are going to start
getting plugged into this z-score formula, or what called on your AP exam formula sheet,
the standardized test statistic. And it doesn’t always say a z-score, because you will be
also learning about something called a t-score, and they don’t simply says x minus mu over
sigma, because you are starting to learn now, in these following chapters, some different
formulas to find the mean, and to find the standard deviation, and those formulas simply
get plugged into this outline that we have. Okay? So, you’re going to be doing a lot
of z-score calculations again, so if you liked that part of statistics, well congratulations,
it’s coming back. Don’t forget that in a z-score calculation setting, that as a continuous
random variable as a mathematical model, and x can take on, again I just said this a second
ago, an infinite number of possibilities. So, when you start seeing z-scores and bell
curves, don’t forget that the probability of x to a specific value is 0, because its
1 over infinity, and infinite number of possibilities, and when you’re doing a continuous random
variable, don’t forget, that the probability that x is greater than the number or less,
and the probability that x is greater than or equal to a number, or it could be in the
other direction, less than or equal to, are going to have exactly the same answer. You
didn’t see that when we did the example about the discrete random variable. Talking
about normality, the definition of normality is again defined by your empirical rule, with
the one standard deviation you have 68% of your data, within two standard deviations
you have 95% of your data, and within three standard deviations, you have 99.7% of your
data. If you have forgotten the graph that your calculator can make for you, automatically
check for normality, then read your book, or go find a past video about a graph called
a normal probability plot, which is where you graph the univariate, quantitative data,
the Xs, are going to be graphed on either the x or y axis, and the other axis, I think
when I did my notes the data was on the x axis, the other axis is going to display the
z-scores. When you put your univariate data on one axis and compare it to the z-scores,
your normal probability plot should be showing a roughly linear pattern, and if it does show
that, then your original data was therefore approximately normal. Let’s see. Don’t
forget that if you, your number summary for a normal distribution is the mean and the
standard deviation, so when you see N(0,1), well the capital N just means that you have
a normal distribution, and if you have a normal distribution with a mean of 0 and a stranded
deviation of 1, that’s the standard normal distribution, that’s the distribution of
z-scores. That’s the z-score table at the front of your textbook that you use every
time you calculate the z-score. So, I verbally reviews quite a few concepts, I hope that
I’ve made some sense out of this stuff, and go do your homework. Mr. Tarrou is out.