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AP.STATS:

VAR‑2 (EU)

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CCSS.Math: You can never have too
much practice dealing with the normal distribution,
because it's really one of those super
important building blocks for the
rest of statistics and really a lot of your life. So what I've done is I've
taken some sample problems. This is from ck12.org
Open Source flex book, their AP Statistics flex book. And I've taken the problems
from their normal distribution chapter. So you could go to their
site and actually look up these same problems. So this first problem, which
of the following data sets is most likely to be
normally distributed? For the other
choices, explain why you believe they would not
follow a normal distribution. So it's a choice. So this is my beliefs
come into play. So this is unusual
in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what
they have here. A, the hand span. Measured from the
tip of the thumb to the tip of the
extended fifth finger. So I think they're
talking about, let me see if I can draw a hand. So that's the index finger. And then you got
the middle finger. And then you got
your ring finger. And you got your pinky. And the hand will look
something like that. I think they're talking
about this distance. From the tip of the
thumb to the tip of the extended
fifth finger, which is a fancy way of saying
the pinky, I think. They're talking about
that distance right there. And they're saying, if
I were to measure it of a random sample of
high school seniors, what would it look like? Well, you know how far this is. This is a combination
of genetics and environmental factors,
maybe how much milk you drank or how much you hung from
your pinky from a bar while you were growing up. So I would think
that it is a sum of a huge number of
random processes. So I would guess that it is
roughly, normally distributed. You know, if I look at my own
hand, and my hand I don't think has grown much since I
was a high school senior. It looks like roughly
nine inches or so. I play guitar. Maybe that helped
me stretch my hand. But it's really
an essay question, so I just have to
say what I feel. So I would guess
that the distribution would look something like this. I don't know. I've never done this. But maybe it has a mean of
eight inches or nine inches and is distributed
something like this. It's distributed
something like that. So maybe it probably does look
like a normal distribution. But probably won't be
a perfect, in fact, I can guarantee you it won't be
a perfect normal distribution. Because one, no one can have
negative length of that span. This distance could
never be negative. So they're going
to have, I guess you could have no hand so that
would maybe be counted as 0. But the distribution wouldn't
go into the negative domain. So it wouldn't be a
perfect normal distribution on the left hand side. It would really
just end here at 0. And even on the right
hand side, there is some physically
impossible hand lengths. No one can have a
hand that's larger than the height
of the atmosphere or an astronomical unit. You would you start
touching the sun. There's some point, which
is physically impossible to get to. And in a true
normal distribution, if I were to flip
a bunch of coins there's some very,
very small probability that I could get a
million heads in a row. It's almost 0. But there's some probability. But in the case of hand span,
there's no way out here, you know, the probability
of human being who happens to be a high
school senior, having a one-mile length
hand span, that's 0. So it's not going to be a
perfect normal distribution of the outliers or as we
get further and further away from the mean. But I think it'll be a pretty
good, in our everyday world as good as we're going
to get, approximation. The normal distribution
is going to be a pretty good approximation for
the distribution that we see. And I guess one thing that
you know-- It's funny. This is high school
seniors, when I did this, it was kind of from my
point of view as a guy. And I would argue that high
school seniors, guys, probably have larger hands than women. So it's possible you actually
have a bimodal distribution. So instead of
having it like this, it's possible that the
distribution looks like this. That you have one peak for
guys, maybe at eight inches. And then maybe a peak for
women at seven inches. And then the distribution
falls off like that. So it's also possible
it could be bimodal. But in general, a
normal distribution is going to be a pretty
good approximation for part A of this problem. Let's see what part B, what
they're asking us to describe. The annual salaries
of all employees of a large shipping company. So if we're talking
about annual salaries, we have minimum
wage laws whatnot. so I would guess that
any corporation, if we're talking about full
time workers at least, there's going to be some
minimum salary that people have. So I would say, and
probably a lot of people will have that minimum
salary because it'll be probably the most
labor intensive jobs, you probably have
most people down there at the low end of the scale. And then you have your
different middle level managers and whatnot. And then you probably
have this big gap. And then you probably have
your true executives, maybe your CEO or whatnot. If this mean, right here,
is maybe $40,000 a year and this is probably
$80,000 where some of the mid-level
managers lie. But this out here, this
will probably be-- Actually if you were to draw it real, the
way I've scaled it right now, this would be about
$200,00, which is actually a reasonable salary for a CEO. But the reality is,
is that this actually might get pushed
way out from there. It might look
something like that. It might be way off the chart. Let's say the CEO made
$5 million in a year because he cashed in a bunch
of options or something. So it would be way over here. And maybe it's a
CEO and a couple of other people, the
CFO or the founders. So my guess is it
definitely wouldn't be a normal distribution. And it would be bimodal. You would have another peak
over here for senior management. Well, if we were
maybe in Europe, this would be
closer to the left. But it won't be a perfect
normal distribution. And you're not going
to have any values below a certain threshold,
below that kind of minimum wage level. So I would call
this, when you have a tail that goes
more to the right than to the left, a right
skewed distribution. Since it has two humps right
here, one there, and one there. We can also says it bimodal. I mean, it depends on what
kind of company this is. But that would be my guess of a
lot of large shipping company's salaries. Let's look at choice
C or problem part C. The annual salaries
of a random sample of 50 CEOs of major companies,
25 women and 25 men. The fact that they
wrote this year, I think they maybe are
implying that maybe men and women, the gender gap
has not been closed fully, and there is some discrepancy. So if I was just purely 50
CEOs of major companies, I would say it's probably
close to a normal distribution. It's probably something
like, once again, there's going to be some level below
which no CEO is willing to work for, although you have
heard of some cases where they work for free. But they're really getting
paid in other ways. If you include all
of those things, there's probably
some base salary that all CEOs make
at least that much. And then it goes up to some
value, the highest probability value. And then it probably has
a long tail to the right. And this is if there
were no gender gap. So this would just be a purely
right skewed distribution where you have a long
tail to the right. Now, if you assume that
there's some gender gap, then you might have
two humps here, which would be a
bimodal distribution. So if you assume
there's some gender gap, this is part C right
here, then maybe there's one hump for women. And if you assume that
women earn less than men, then another hump for men. And there are 25 of
each so there wouldn't necessarily more men than women. And then it would skew all
the way off to the right. And in fact, I think there
would probably be chance that you have this other
notion here where you have these super CEOs, or mega
CEOs who make millions, while most CEOs
probably just make, I'll put it in quotation
marks, "a few hundred thousand dollars" while there's
a small subset that are way off many standard
deviations to the right. So it could even be a
trimodal distribution here. So that's choice C. And
then so far, choice A looks like the best
candidate for a pure, or the closest to being
a normal distribution. Let's see what D is. The date of 100 pennies
taken from a cash drawer in a convenience store. 100 pennies. So that's actually an
interesting experiment. But I would guess,
and once again, this is really a question
where I get to express my feelings about these things. As long as your
answer is reasonable, I would say that it is right. Most pennies are newer pennies. Because they go
out of commission. They get traded out. They get worn out as they age. They get lost, or they get
pressed at the little tourist place into those
little souvenir things. I'm not even sure
if that's legal, if you could do that
the money legally. So my guess is that if
you were to plot it, you would have a
ton of pennies that are, within the last few years,
So the date of 100 pennies, not their age, so the dates,
so if this is 2010, I would guess that
right now, you're not going to find
any 2010 pennies. But you're probably going to
find a ton of 2009 pennies. And then it probably just
goes down from there. And of course, you're
not going to find pennies that are older than
the United States or before they even
started printing pennies. So it's obvious this tail
isn't going to go to the left forever. But my guess is
you're going to have a left skewed distribution. Where you have the bulk of
the distribution on the right, but the tail goes
off to the left. That's why it's called a
left skewed distribution. Sometimes this is called a
negatively skewed distribution. And similarly, this right skewed
distribution, or this right you distribution, sometimes is
called positively skewed. And if you have only one hump. You don't have a multimodal
distribution like this, in a left skewed
distribution, your mean is going to be to the
left of your median. So in this case,
maybe your median might be someplace over here. But since you have this
long tail to the left, your mean might be
someplace over here. And likewise, in
this distribution, your median, your middle value,
might be some place like this. But because it's right skewed,
and for the most part only has one big hump. this hump won't change things
too much because it's small, your mean is going to
be to the right of it. So that's another
reason why it's called a right skewed or
positively skewed distribution. So to answer the
question, you know, these are my feelings
about all of them. But I would say--
the other choices explain why you believe
they would not follow-- or they said, which of
the following data sets is most likely to be
normally distributed. Well, I would say choice
A. But it's really a matter of opinion, at
least in this question.