Inferential Statistics - Single sample tests Video Lecture | What is Data Analytics? - Data & Analytics

in our previous lecture we spoke about
the need for we try to motivate the need
for inferential statistics through the
context of hypothesis testing so we
spoke about why we needed it where it
would apply and so on and we concluded
that lecture by coming up with the
template coming up with the rubric
essentially of what it is that one needs
to do with hypothesis testing so we
started off with instructions like you
need a null hypothesis and alternate
hypothesis and the whole thing was
fairly general so in today's lecture
what we're going to do is today's
lecture continues our focus on
hypothesis tests and we're going to talk
about something called single sample
tests in the last class you were seen
that we gave two sets of examples we
gave examples of single sample tests and
two sample tests and today we're going
to focus on single sample tests and what
we're going to do is we're going to
illustrate that template that you saw
with by illustrating one test so the
test is going to be the single sample
z-test and we're going to show you the
mechanics behind it and kind of give you
the reason why we do some of the math
that we do and and then we will talk
about some of the other tests that are
there as well okay so the single sample
z-test is a test that is used when you
want to make some inferences about the
population mean note that you know again
the clarity here is that you're not
saying you're using the sample but
you're not interested in just reporting
the sample mean which is what you might
have done with descriptive statistics
but you're interested in ultimately
making a statement about the population
mean okay and this is a test where the
you need to know the variance of you can
think of it as variance or standard
deviation but you need to know
this variance of the population and then
that's a requirement so it's not the
same thing as computing the variance
from the sample so that's called the
sample variance and there's a formula
for that there's actual data but this is
more useful when there is bin historic
data and you actually know the
population variance but ultimately
you're doing a test to see if the
population you're doing a test about the
population mean so and another case
where this test finds an application is
when the sample size is fairly large so
this is the one exception to the this
rule that you need to know the variance
in some cases when the sample size is
very large and larger gets defined you
know by approximately 30 so that's the
magic number that people use here and
when your sample size is 30 or larger
you consider you reason that you have a
large enough sample and in that one
instance you don't need to know the
variance you can actually calculate the
sample variance from the data itself and
use that for this test so just to
quickly summarize with the single sample
z-test you have a single sample and then
you're testing you're making some
inferences about the population mean
based off of this sample and typically
in a single sample z-test you you need
to know the variance with the x mall
exception that you can also use the
single sample z-test if you have a large
enough sample size and you don't know
the variance okay so the example we're
going to use to motivate this test is
one that you've already seen in the
previous class so we spoke about this
problem
on average phosphate levels in blood
and we said that we created I want to be
very clear we created this kind of a
medical scenario I'm not I'm not saying
this is medically accurate we are
interested in the statistics of it so we
imagine there's this doctor or this this
Public Health System which says that
your average phosphate the average
phosphate level in your blood should be
less than four point eight milligrams
per deciliter and the idea here is that
doctors or whoever understand that not
every time so you take a blood reading
you take your blood and you take a
reading of the phosphate level not every
time is it going to be four point eight
or less sometimes it might be more
sometimes it might be less but the whole
idea is you're trying to see if the
average is less and when you say I'm
interested in the average you're not
interested in the average of some sample
that you've taken and here a sample
would be if you took five blood tests
perhaps different machines give
different results perhaps different
times of the day give different results
perhaps what you ate in the morning
affects it but the whole point is you
just have that sample at hand
but you are interested in saying
something about the population so let's
let's just step back and you know go
through each of these bullet points one
more time in the context of this example
so we said what are we testing I'm
interested in the population mean so you
might ask yourself in this particular
example what is the population right and
you can think of the population in a
couple of different ways you can think
of it as a concept of what your true
average and true distribution is so so
let's say that yes you can take a blood
test and when you take a blood test it's
like you're taking a sample but there's
the concept of what of what the
phosphate is in your blood at all times
right so doing a test is gives you one
peek into the reality but there exists
this concept of reality which is that
there's some distribution
and we are assuming that distribution
doesn't change over time and you know
that distribution has a certain mean and
we're very interested in this mean and
this is the distribution of the
phosphate levels in your blood this is
that reality which we don't know and you
know you can think of it as this this
Oracle somewhere that knows what your
true phosphate levels in blood is at all
times in its some distribution and at
any point of time you go take a blood
test it's like you're getting a random
you're getting a number from this random
variable which is in the form of a
distribution and you're very interested
in the mean of the distribution so
that's the population the population is
your true phosphate levels in your blood
at any point of time and so it's a
concept and if you kind of like to think
of populations as as actual data points
another way you can think of this is to
say the population is what is this data
set this very large data set that you
would see if you were to continuously
take blood tests infinite number of
times with infinite number of machines
you know over multiple days or whatever
so so you can think of constructing as
population in your head as a very large
data set but the truth is ultimately you
never you never have the data set or you
don't otherwise we wouldn't be doing
inferential statistics but what you do
have is a sample that sample can be of
some size we've not discussed that it
can be 5 10 20 30 40 data points and the
idea here is that you know the variance
so I've said that it's a known standard
deviation of 0.4 milligrams per
deciliter so just mind you that this is
the standard deviation not of the data
set of the samples that you got but it
is the standard deviation of the
population so it's like this there's
this population distribution you're
trying to answer some questions about
it's mean but someone somehow you know
what the standard deviation of this
distribution is you know someone
whispered it to you or you know you know
it from fundamental principles or you
might reason that historically it's been
equal to this value and should not have
changed but you know the standard
deviation in this particular example and
we've already talked about how if you
have a large enough sample size you
don't actually need to know the standard
deviation you can compute it okay so
here is the data right so I've told you
that we know the standard deviation but
you also this out here is the data and I
have explicitly not given you the exact
data point so just to give you an idea
you each of these data points comes from
you know going and doing a blood test so
you did a blood test and you got four
point one you did it again you got three
point nine and so on and there's this
list and this is what we call as the
sample okay so what do you do with this
data how do you conduct the test let's
go back to this rubric that we created
that's just so it's like this template
that we discussed in the end of last
class and do to conduct any hypothesis
test
so the first bullet point says have a
null and an alternate hypothesis and
that's what we are going to do the null
hypothesis here is to say that mu naught
which means which refers to the
population mean and you can use the
abused mu naught here you can use mu as
well that's also you might see textbooks
do that but the idea is that the null
hypothesis says that the true mean of
the population is less than 4.8 I don't
know the answer to this question that's
what I'm hypothesizing we are going to
do some you know mechanics we're going
to go through some process at the end of
it we're going to see if the null
hypothesis is true or not
colloquially speaking so your null
hypothesis like we said
de naranja alternate together need to be
mutually exclusive collectively
exhaustive um you not says that Yin
Allah is less than 4.8 that's your null
hypothesis so the alternate hypothesis
should be that mu naught is greater than
4.8 okay the next step we said is do
some basic calculations arithmetic on
the data to create a single number
called the test statistic and the math
that we're going to be doing here is
fairly straightforward
we're going to take X bar which means
the sample mean okay
the sample mean here would just mean
that you take these data points out here
and you take their average so you take
all the data points that you've
collected and you take their average so
we do that and we then calculate X bar
minus mu naught and here mu naught is
4.8 it's the number that you're
hypothesizing and you divide that by
Sigma divided by square root of n so
here Sigma refers to that standard
deviation that you already know right so
this is the point four that we spoke
about occur so we are already given that
the standard deviation is point four
okay so given this we compute X bar
minus mu naught so X bar is the average
of the sample mu naught is the four
point eight the number that you are
hypothesizing Sigma is the standard
deviation that you're given and n is the
number of data points in that sample so
if your sample size is 1015 you would
substitute ten or fifteen okay so that's
how you calculate something called the Z
statistic so why are you calculating
this value we've said in the next bullet
point that if we assume the null
hypothesis to be true and make some
assumptions about distributions I won't
say that every time but if we assume the
null hypothesis to be true then
technically the test statistic should be
no different
then pulling a random number from a
specific probability distribution so if
that the whole idea is if the test
statistic if the null hypothesis is true
then this test statistic Z stat right
should be equivalent calculating Z stat
so you can plug in numbers for X bar mu
mu naught Sigma and N and you will
calculate a z stat right once you
calculate the z stat if the null
hypothesis is true then the Z stat that
you're getting should be equivalent
should be the same thing as pulling a
random number out of a specific
probability distribution what is the
specific probability distribution in the
case of the single sample z-test this
distribution is called the Z
distribution okay and the Z distribution
is nothing but a normal distribution
with mean of 0 so I've used this
nomenclature discussed how this is
standard nomenclature but the N here
means it's a normal distribution with a
mean of 0 which is the first number that
you see so you have a normal
distribution with a mean of 0 right and
a standard deviation of 1 and and one
square is a convenient way to represent
it because you know very clearly that
you're talking about 1 refers to the
standard deviation and 1 square is also
equal to 1 so the standard deviation or
the variance is equal to 1 and so this
is what is known as a Z distribution so
we are saying that if the null
hypothesis is true the Z statistic that
you compute with this data should be the
same or should be equivalent to pulling
a random number from a Z distribution ok
this is a very useful thing to say and
what we're going to do is we're going to
come back to what we do next because of
the statement but before we do that I
want to make sure that you understand
why if the null hypothesis is
that the z-statistic computing a Z
statistic from the data is equivalent to
pulling a random number from a
distribution that is normally
distributed with mean 0 and standard
deviation 1 so let's let's go to the
next slide to kind of do that okay so at
the start you had this distribution
right so this is this distribution is
the population distribution so this is
the population okay
the population if the null hypothesis is
true would have a mean equals equaling
mu correct I'm using mu and mu not
little interchangeably out here but I
don't want you to get confused by that
but the idea is whatever their
hypothesis is if your null hypothesis is
true then the mean of this distribution
is equal to 4.8 technically it's less
than or equal to 4.8 but we are going to
take the extreme case so we are going to
come to one end of it and say it's equal
to 4.8 okay so this is the extreme
situation where the mean is actually
equal to 4.8 and we are already given
the standard deviation right so you've
already been told that Sigma is 0.4 and
you're given that value so first let
this is just the distribution of the
population so you've built the
distribution of the population now when
you go take a sample from this
population of some size n what do you
get okay what you get is as we've
discussed if I take a sample of five
data points six data points and then
compute a mean from that sample we know
that then that arithmetic mean that you
compute from a sample need not always be
equal to the MU exactly sometimes it's
mu sometimes it's a little higher than
mu sometimes it's a little lower than me
but we've already discussed as to how
that is also a distribution and that
distribution called the sampling
distribution is also normally
distributed with the same mean mu but
with a standard deviation of Sigma by
square root of n where n is the number
of samples you took to compute that mean
so if you took five data so if you if
you technically just take one sample and
compute the mean from it you'll get the
same distribution again get the same
original distribution so if you
substitute n is equal to one nothing
changes and that should be intuitive if
your sample size is just one data point
and your computing the average of one
data point it's literally like you're
recreating the distribution if that n is
goes to infinity then your variance goes
to zero right so as your sample size
keeps on increasing you're literally
going to be sitting on top of this line
and you really won't have this
distribution but for all finite finite
sizes of samples what you have is a
distribution for samples which is
normally distributed with mean mu and
standard deviation Sigma by square root
of n ok now what we're going to do now
mind you this is the distribution of
sample means so in some sense this is
the distribution of x-bar okay now what
we're going to do is we're going to
subtract mu from this distribution from
this distribution we're going to
subtract mu okay what effect does that
have from a distribution from a normal
distribution essentially if you just
subtract a number right
it's literally like just shifting the
distribution and centering it so because
you're just subtracting a number you're
not affecting the standard deviation so
this gets unaffected but when you just
subtract a number from the distribution
you can think of subtracting from a
distribution as you take each data point
and then subtract the same number from
it but you can think of it as computing
that x-bar and then subtracting it but
in either case it has the effect of just
shifting the distribution it has a
effect of centering the distribution at
another location and that's what happens
when you subtract mu like when you take
mu out of the x-bar right so it's this
what I've done I've taken MU out of
x-bar it has the effect of moving this
distribution to another location which
is now centered around the mean equal to
zero okay
now what happens if you then take this
distribution and divide it by the
standard deviation so what happens if
you divide it by this number Sigma by
square root of n okay the effect that
that has is in rescaling the standard
deviation such that you now have a
distribution which is normally
distributed when you divide right
something that is already centered
around zero so the distribution is
already centered around zero which means
it has some positive value some negative
values right and now you go divided by a
number the effect of the division is
that because it's already centered
around zero it's not actually going to
change the central location of the
distribution
it's instead only going to widen it or
narrow it depending on what you divide
it by so I've kind of shown the
distribution getting narrower but that
need not be the case the distribution
could have just gotten wider it just
depends on whether Sigma by square root
of n was greater than or less than one
so if it's greater than one then you
would by dividing by Sigma by square
root of n you would be making the
distribution more narrow but if Sigma by
square root of one was smaller than one
then it would have the effect of
widening the distribution but ultimately
when you go divided the normal
distribution which is already centered
around zero all you're doing is you're
either stretching the distribution or
kind of crunching it you can think of it
as scaling it and now you have a
standard deviation of one so that's how
you get the whole idea of X so so the
first step we took X bar right and then
we subtracted the MU from it that's
where we got this value and now after
dividing by this number Sigma by square
root of n you went and divided this by
this number to get your normal
distribution with mean 0 and standard
deviation 1 which is what we said was
the Z distribution okay
God perfect so now what we're going to
do is go back to this rubric okay so
great so we've reasoned that you have to
calculate the Z stat and we've already
said that the Z stat if the null
hypothesis is true should be like
pulling a random number from a normal
distribution what we're going to do now
is we're going to test the probability
that this statistic that you got we're
going to say okay if your null
hypothesis was true I should have gotten
a random number from here right but
let's actually look at the actual number
that I got right does my Z stat actually
look like it could be something that I
could have pulled out of a normal 0
comma 1 square because if it does not
right then something that I assumed was
wrong right I said that this Z stat
should look like something that I pulled
out of a normal 0 comma 1 square if the
null hypothesis is true so let me go
take an actual look at the Z stat number
and if it looks like it could not have
come from this distribution if it looks
like it was unlikely to have come from
this distribution then I can reason that
maybe the null hypothesis was not true
and I can reject the null hypothesis so
that's what we're going to do with the
next step the next step is to take the Z
stat okay and plug it in to this normal
0 comma 1 square and see how extreme is
this actual number given that we know
the normal 0 comma 1 square and so we
know the potential values it can take
for instance I've already told you it's
a normal distribution with mean 0 comma
1 standard deviation so from this
distribution if I were to pull a random
number how likely is it that I would see
a number like 55 right
that's too high the standard deviation
is 1 the mean is 0 it's almost
impossible that you would pull a number
like 55 Oh
or minus 20 so for that matter right so
if it turns out that the Z stat is too
extreme a value to be coming from this
normal distribution then we can maybe
make a statement about the null
hypothesis so that's what we're going to
do as a next step yeah let me just clean
this up okay so I've just restated the
official definition what we're going to
be calculating is something called a
p-value and this is the probability of
seeing a test statistic as Extreme as
the calculated value if the null
hypothesis is true with the core idea
being that if it looks too extreme that
if the p-value of the probability of
seeing this test statistic is really low
then perhaps the null hypothesis was not
true to start with so for instance out
here if the Z statistic you computed was
1.2 right so I just I'm just giving you
a number to go go with then the core
idea is that you would calculate a
p-value based on the standard null
hypothesis which is that mu naught is
less than 4.8 and you would say let me
take this distribution which is the Z
distribution or the normal 0 comma 1
square okay so this is the normal 0
comma 1 square and I'm going to go place
1.2 here and I'm going to compute a
probability and this the probability is
a probability to the right side of 1.2
it's it's the area under the curve out
here and the idea is because you you are
then quantifying the probability of
seeing something as Extreme as 1.2 or
greater is equal to the area under this
curve and that might happen to be any
value so I mean I think in this case it
happens to be something like 13% or
whatever but if this number was really
low
number was point zero zero one you might
say look this number is so low that I'm
going to reject the null hypothesis and
if you cannot find something that's that
extreme the standard thing that you do
is you fail to reject the null
hypothesis you technically never accept
the null hypothesis so so that's the
core idea and you know you can you can
also think of this in a couple of other
ways for instance if your null
hypothesis were that mu is greater than
four point eight right then you would be
looking at the area to the left of your
curve but typically in a situation like
that you actually want to the the
statistical test so it's not common to
CP values greater than 0.5 because at
that point you start by saying look I've
computed a Z statistics that's already
positive that's one point two and so the
I know even before I go put this line
out here that I'm going to get a
probability greater than 0.5 right so
for instance your z stat was computed to
be exactly equal to zero you know that
if your null hypothesis was mu is
greater than 4.8 that it will be 0.5 so
any z stat even greater than that is
bound to be greater than 0.5 when you
don't need statistics for that but an
but another interesting situation which
a lot of people work with is called the
two-tailed case these two are called one
tail one tail okay so you also have the
two-tailed case where your null
hypothesis is really that mu is equal to
four point eight and you're interested
in rejecting the null hypothesis whether
that mu because is too large
meaning it's large enough that you can
you know say that it can't be equal to
four point eight right or if it's small
enough so you're happy to reject if you
see ever
means that shows that mu cannot be 4.8
on either account maybe it's because the
data suggests that it's too large or
maybe because the data suggests that
it's too small and that's called the
two-tailed case now there's lots of
different software where many of the
steps that we've done can be actually
taken care of and it doesn't take much
even a simple excel sheet if you just go
dump the data and say do Izzie test it
and do it but somewhere understanding
the mechanics of this and getting it to
the stage of the Z statistic at least
you know brings about some sense of
control and transparency in your
understanding but once you get to this
stage computing this area right whether
it's on the left hand side right hand
side or either side can be done fairly
it's it's not something that can easily
be done by hand so what our textbooks do
is if you've taken most statistics
textbooks will have these pages towards
the end and then they've given in the
form of tables so you can take the Z
statistic the number that you computed
and go plug that in to this table and it
will tell you the probabilities right
and usually there'll be a diagram on top
to hint whether they're giving you the
probabilities to the left-hand side or
the right-hand side or both sides and
you know as long as you know what
they're giving you it's fairly easy to
figure out figure out whatever it is
that you need so if you want the
right-hand side but they giving you only
the left-hand side then you can just
subtract subtract the number they're
giving you from one because you know the
total area under the curve is equal to
one okay but what I'm going to do is I'm
just going to give you some simple Excel
functions that do this for you
for instance in Excel you can just the
convention is to give you the area to
the left hand side so for instance what
I do here is I subtract that from one
and the norm s dist
is what refers to the Z distribution and
the true refers to the fact that I'm not
interested in just the high
I need the area under the curve so
that's what that does so for the for the
right-hand side tail you can use this
for the left-hand side tail you can use
this a simple multiplication by two with
the left hand side case gives you the
two-tailed situation okay so with this
we've discussed in greater detail the
single sample z-test so now let's look
at a couple of other single sample tests
what would I provide you with here is
the list of them and the formulas and
I'll give you some idea of the context
in which they have used but we won't
derive it or go through it in the same
detail as the Z test
so we've already discussed the Z test
let's now look at the next test which is
the T test okay so we finished with the
Z test so now we're going to look at the
T test okay so with the T test it's it's
a very useful test and it also tries to
test this it essentially tries to do the
same job the Z test is doing which is to
make some statement about the population
mean
so same same problem statement in some
sense the one big difference is you're
not given the variance and in most
situations in life and in statistics you
won't know the variance I mean just
think about how how fairly unrealistic
it is that you already know the variance
of the population in a situation where
you're trying to make a statement about
the mean I mean the only reason you're
doing this test is because you don't
know what the population mean is so
you're trying to you're making a
hypothesis you're taking a sample and
then you're testing that sample you're
working with that sample to make a
statement about the population mean so
this i mean think of it as there's some
uncertainty about the population mean in
the first place and that's why you're
doing this test to assume that in such a
situation you already know the
population variance is not very need not
be very realistic so this test works the
same way so if you look at it it's got
the same x-bar it's got the same mu it's
got the same square root of n but this S
is different from this Sigma
okay and the difference is here Sigma
was given in the Z test so in the Z test
Sigma is given but in this test the S is
computed it's computed from the data so
you actually go back to the sample data
and you calculate the variance or
standard deviation from the data using
the same formula for dispersion that we
would have discussed when we spoke about
standard deviation and descriptive
statistics using the N minus 1 idea and
if you if you don't remember you can go
back and see that lecture but the idea
is that you compute the standard
deviation and you plug that value in to
get the T distribution now a couple of
things that are worth noting is that we
spoke about how if you know the variance
you can use the Z distribution if you
don't know the variance you can use the
T distribution but there is this
exception right we said if your sample
size is large enough then you can
technically use the Z distribution and
just compute the variance and consider
the variance to be the truth consider
the variance to be the Sigma and go
ahead I personally find that a little
confusing I think the the that's fine if
you know that's that's what's there in
text and that's what people do and
that's a reasonable approximation but
but the point is you can't you cannot go
wrong with using the t-test when you
don't have the variance so even if you
have a large enough sample size and the
idea is that the T distribution becomes
it approximates the Z distribution quite
well when your sample size is greater
than 30 there for all practical purposes
the T distribution looks exactly like
the Z distribution but but to keep
things really simple you can just follow
the simple rule that you don't know if
you know the variance just use the Z if
you don't know the variance just use the
T and and that should let just keep you
clear the other thing to mention out
here
is that this DOF or degrees of freedom
we've mentioned that out here we briefly
spoke about that concept when we were
talking again about standard deviation
without going into too much detail into
degrees of freedom again the simple
thing to keep in mind is that the T
distribution is not one distribution
right I mean it's one distribution but
in the sense that the T distribution you
can think of the degrees of freedom as a
parameter that goes with the T
distribution so just like the normal
distribution if you say the normal
distribution you need to mention the
mean and the variance for you to have a
to actually draw the exact distribution
or to do some computation on it it's no
point coming to someone and saying how
likely is it to see a one point two in a
normal distribution right that that
question doesn't make sense a normal
distribution with what mean and what
variance and then I can answer your
question you can think of degrees of
freedom in a similar light which is that
the T distribution itself is not
completely defined until I mentioned to
you what the degrees of freedom are so T
distribution with three degrees of
degrees of freedom equal to three looks
different from a T distribution with
degrees of freedom 4 and the core idea
that you need to know is that the T
distribution has a mean of zero okay and
it looks very similar to the normal
distribution of mean 0 and standard
deviation one with the exception that as
the degrees of freedom keep increasing
so when you go two degrees of freedom of
30 and greater and at some point it it's
exactly the normal distribution so the T
distribution with a large enough degrees
of freedom is exactly like the normal
distribution with mean 0 and standard
deviation 1 but as the degrees of
freedom keep decreasing and come all the
way down to the lowest degrees of
freedom you can have is 1 when it comes
all the way down to degrees of freedom
equal to 1 you'll find that it still
looks a little bit like the normal
distribution with mean 0 and standard
deviation 1 but it's a little shorter
is shorter in the center and has fatter
tails on the sides and so that's how it
deviates from the normal distribution
but but all that you need to know is
that the degrees of freedom get defined
by the concept of n minus 1 so number of
data points minus 1 tells you the
degrees of freedom and once you know the
degrees of freedom you know which T
distribution to look up in the table so
you know how to draw the curve and then
calculate probabilities from it again
Excel uses Excel has some slightly nicer
functions for it so if you're just
interested in looking at the left-hand
side of the distribution you just use T
dist T dot lists if you're interested in
T dot and the right-hand side you do T
dot this dot R T or on both sides 0 T
dot dist 2 T so you don't need to
actually do the 1 minus and so on that
we were talking with the Z distribution
Excel already has some inbuilt functions
to just point to which side of the
distribution you are interested in ok so
we go now to the next we finish with the
T distribution we go now next to the
next test which is the chi-square test
and the chi-square test has a couple of
different types of tests but the one
that we are interested in now is the
chi-square test for variance I'm using
the words variance and standard
deviation a little interchangeably one
is just the square of the other the test
is ultimately one for variance and and
if you're testing variance are
essentially testing standard deviation
so if it's easy for you to think
standard deviation you can keep that in
mind and an example for instance of the
chi-square test is you're really
interested in looking at a sample but
you're not interested in making a
statement about the population mean
you're instead interested in making a
statement about the population variance
so you're interested in saying is the
pop just like in the Z test and the
t-test you were interested in saying
something like is the population mean
equal to 4.8 or is the population mean
less than 4.8 similarly here you'd be
interested in saying things like is the
population variance equal to 0.5 0.3
whatever whatever number you have in
mind
the important thing is you have a number
in mind and you're trying to see if the
sample that you're taking with the
sample that you're taking can you see
something about the population variance
being equal to this magical number that
you have in your head okay and the
mechanics of the test is fairly
straightforward and it's here the Sigma
naught is the hypothesized variance so
this is the number that you want to
compare it to this is the equivalent of
the 4.8 that was there for means the s
square is the sample is the variance
that you compute from the data from the
sample right you take that you take the
data set of the samples and you compute
a standard deviation you compute a
variance from that and that's s Square
and the way you do that again to remind
you is this that 1 by n minus 1 in the
formula for the calculation of standard
deviation you would be using that and
and and that's how you calculate this
test statistic which then come gets
compared to a chi-square distribution
with n minus 1 degrees of freedom just
like in the first case it get it got
compared to a Z distribution and this
got compared to it the the T statistic
got compared to a t-distribution this
the same way the chi-square distribution
gets compared to a chi-square
distribution
agreat so a couple of things to note is
that if you again chi-square also uses
the concept of degrees of freedom so
think of the degrees of freedom as
something that defines the exact
distribution you're interested in
because a chi-square distribution with
three degrees of freedom is a different
distribution than a chi-square it's a
different density function right it
looks different it has different
mathematical properties then a
chi-square distribution with you know 4
degrees of freedom 5 degrees of freedom
so so the degrees of freedom help you
define the exact distribution and its
parameters and and it's you know mean
variance and so on
but essentially that's what you would
use are you you would need to use the
degrees of freedom and that's also the
same as before it's n minus one so
number of data points minus one okay and
chi-square with Excel out here just uses
chi-square distillate
side and chi-square discard the as the
right side I don't see them having
something for two tailed
but but I might be wrong but as long as
you know you have these two you can
quite easily just draw the graph in your
head and figure out which side if you're
interested in a two tailed distribution
how you would compute that yeah great so
we finally come to our last single
sample test which is called the
proportion Z test and the idea here is
you're testing something that's a
proportion so you're testing something
like if you are given a so so you're
testing a hypothesis like less than 30%
of the shoppers who come to my online
store are women so you can say again we
can go the two-tailed way or you can go
the one-tailed way you can say less than
30% you can say a 30% of my shoppers in
my online store or women or you can say
greater than 30% of the shoppers moment
the key out here is that whatever sample
you collect to actually test this
hypothesis the hypothesis so let's fix
on the hypothesis let's say the
hypothesis is less than or equal to 30
percent of the shoppers who come to my
online store are women okay the idea is
like all hypothesis testing you will now
you have this hypothesis you will now go
and collect a sample the sample in this
particular example could be something
like you actually give a survey at the
end of the purchase or something and
people actually say that the main of
mean so you collect a sample how does
the sample look the answer is that the
sample unlike in the previous examples
where you would have seen an actual
number right so you in the previous
example the phosphate examples you saw
numbers like 4.1 3.5 these were actual
readings from blood tests here you're
going to get something that's binary
right the person is either going to say
that either female or not so it's a
series of ones and zeros very similar to
the idea behind Bernoulli trials and
what you're doing is you're now looking
at the sample data of ones and zeros
which and then answering that question
of whether and then saying something
about the hypothesis which is that less
than 30% of the people who come to my
shop are women and you know this has the
same intuition as all the other forms of
inferential statistics which is you know
if you if for instance you take a
hundred samples and you know all hundred
of them point to the the the shoppers
being women then you know you're likely
to reject the idea that only less than
30% of the shoppers who come to my
online store are women okay but the idea
is if you notice something this this
looks in every way shape and form like
the Bernoulli distribution and the fact
that you're counting how many right so
the Bernoulli distribution had to do
with probability of heads or tails so
that would be probability of it being
male or female but if you're interested
in out of a size of n people who arrive
how many of them sort of hundred people
who arrive are is it less than thirty
who are women that ties this to the
binomial distribution right and yet you
see you don't see the binomial
distribution being used in the test you
instead see the same idea of Z right so
yet again calculating a Z statistic with
this test and you're comparing it to the
Z distribution which is normal right
which is normal zero comma one square
and the idea here is fairly simple if
you remember we spoke about the binomial
you're approximating this binomial
distribution to a normal distribution
that's that's the right way to put it
and and that's what you're doing or
chill and it should also you know be
intuitive in that the P hat out here is
the calculated proportion so you take a
data set let's say you took 30 people or
40 people or 50 people who came to your
store and you actually found that
exactly 23 of those 50 were females so
that proportion is what the proportion
that you get from the sample is what you
have is P hat P naught is the
hypothesized proportion so P naught
would be the 30 percent so this would be
you know sample right and this is the
equivalent of mu naught so this is the
population a proportion the the number
that you're hypothesizing and of course
is the sample size okay and and if you
look at it this also looks very much
like that X bar minus mu concept and in
some way this formula out here in the
denominator should remind you of the
formula that you saw for standard
deviation in the binomial distribution
class so you're doing something very
similar to the X X bar minus mu by Sigma
over square root of n it's in construct
identical to that but you're using the
binomial distribution for calculating
things like the variance but you are
also saying hey this I believe should be
you can approximate it to the normal
distribution so I'm just going to use
the Z distribution to calculate my
p-values or I'm going to use the Z
tables okay just in quick conclusion
just going back to the rubric that we
created I'll just clean this up you know
the final idea is that you use these Z
tables you can use you know Z tables by
by Z tables I mean like you can use the
back of a statistics textbook you can
use Excel or MATLAB or are just
important that you know how to do it
with at least one of these softwares and
the core idea is that if you get a low
enough p value all of these help you
and calculate a p-value or a probability
and if you get a low enough p-value you
can use that as grounds for rejecting
the null hypothesis and saying I reject
the hypothesis that mu naught is less
than or equal to four point eight and
also just keep in mind on the flip side
you never say I accept the null
hypothesis and and and you you can you
can only say that I you fail to reject
the null hypothesis okay
I hope that clarified the use of gave
you one illustrative example of the
single sample test and the idea behind
the mechanics of it and introduced you
to the other tests and in the next class
what we're going to do is we're going to
talk about two sample tests and we're
going to go also beyond that we're going
to talk about the idea of having
multiple samples okay thank you