Hypotheis Test for a Mean from a Small Sample Using StatCrunch Video Lecture | Statcrunch Tutorials for Beginners - Data & Analytics

the mean waiting time at the
drive-through of a fast-food restaurant
from the time and order is placed to the
time the orders received as 84.9 seconds
the manager devises a new drive-through
system that he believes will decrease
wait time as a test he initiates the new
system at his restaurant and measures
the wait time for 10 randomly selected
orders the wait times are provided in
the table now because the sample size is
small the manager must verify that week
time is normally distributed and sample
does not contain any outliers in fact
even before that the manager has to make
sure that this data was randomly
selected and so you see on the obtained
10 randomly selected orders so we're
going to treat these 10 randomly
selected orders as a simple random
sample okay so again we have a small
sample size so we need to verify two
things one we need to verify that the
data come from a population that's
normally distributed second that the
data have no outliers we verify the
normal distribution requirement via a
normal probability plot and you can see
the normal probability plot on the
screen let me zoom it in a little bit
and notice that all the dots lie within
the confidence bands because all the
dots line is in the confidence band it's
reasonable to conclude that this data
comes from a population that's at least
approximately normal we then want to
verify that there are no outliers we do
that with the box plot remember in a box
plot if there are outliers they're going
to show up as little asterisks right
outside them out in the tails so there
are no outliers in this data set and in
fact we can see the median is a little
bit
than 80 see that so this might be some
visual evidence to suggest hey maybe
wait times did go down but we want to
know whether or not any the mean wait
time of these 10 is significantly less
than 80 4.91 last modern requirement is
the independence requirement we have ten
randomly selected orders we've got to be
at least a thousand rate orders placed
at any given day we'd want customers a
thousand customers and what's five
percent of even a thousand fifty so our
sample size is less than five percent of
whatever our population sizes you would
think that you'd have to have at least a
thousand customers to stay in business
right so we're probably good to go on
that model requirement as well so yes
the conditions are satisfied now Part B
is asking is the new system effective is
the new system effective and for that
question we're going to go ahead and go
through the entire hypothesis test
procedure and so to go through the
entire hypothesis test procedure to see
if the system is effective first thing I
need to do is establish my null and
alternative hypothesis again the null
hypothesis a statement of no change
there you go the hype null hypothesis is
always a statement of no change no
difference so what's the statement of no
change no difference here
yeah that the mean with the new system
is the same as the mean with the old
system mean is still 80 4.9 seconds what
are we trying to gather evidence to show
we want to know if the mean times have
they have decreased so we want to know
if the mean is less than 80 4.9 seconds
i mean we wouldn't want to know if the
mean is changed because we don't want
the means to go up we want the mean
weight time to go down the next step is
to determine the level of significance
what's our level of significance I think
it was 0 point 0 1 so this in this test
we're having a very low level of
significance which means that we want to
avoid making a type 1 error we want to
avoid instituting a new policy if the
system really isn't effective system
really doesn't reduce wait times right
because the lower the level of
significance the less likely it is that
you make a type one error and remember
type one error means rejecting the null
when the null is true in other words
saying week times have decreased when in
fact they have not step three we want to
get the p-value and we're going to get
the p value from statcrunch technology
right so the nice thing about my stat
lab is I can just open this in
statcrunch right I have statcrunch
already opens don't I so i'll just copy
this to the clipboard
over to statcrunch
and paste it in there we go very nice
and this is wait Jess
so to get the p-value I go to stat T
stats one sample and now i have with
data i have wait times is my variable my
null is that the mean is 84.9 that was
the mean stated in an all right that's
the wait time historically that we've
had and my alternative is a less than
alternatives i'm looking together
against the show that wait times have
decreased now i click calculate and I
get a p-value of point zero seven p
value of point zero seven it would be
nice to be able to interpret this
p-value this p-value means that if we
were to conduct this experiment a
hundred different times we would expect
to get a sample mean of 78 or less in
about seven of those experiments if the
true population mean was actually 80 4.9
seconds and so now the question that you
have to ask yourself is are these
results that we got unusual if we're
going to assume the mean is still 80 4.9
seconds in other words you say wow i
just observe something that happens only
seven times out of 100 it must mean that
this new system is affected that's
that's how you would answer if you
weren't given a specific level of
significance to compare your p-value
against we happen to be given a specific
p value and for us the p-value is
greater than the level of significance
because the p-value is point zero seven
and the level of significance is point
zero one and when the p-value is greater
than the level of significance what's
our conclusion do not reject the
statement in the null hypothesis so we
would conclude
food there is not sufficient evidence to
suggest the mean weight tile has
decreased with
the new drive-through system
so we're we're saying we're not going to
go ahead and institute this new approach
this new policy now let's say that the
manager is all disappointed in these
results and says ooh I really wanted to
institute this policy I'm just going to
go ahead and change my level of
significance 2.1 would that be fair or
to do is that something that's allowed
change your level of significance after
the fact after you've collected your
data and run your test no that would be
sort of cheating right that would be
cheating so to speak so that's why it's
important to choose your level of
significance prior to the data
collection process as opposed to
subsequent to it