Find area under the normal curve; find normal scores using StatCrunch Video Lecture | Statcrunch Tutorials for Beginners - Data & Analytics

the time required for an automotive
center to complete an oil change service
and an automobile approximately follows
a normal distribution with a mean of 19
minutes and a standard deviation of 2
minutes I'm home center guarantees the
customer customers that the service will
take no longer than 20 minutes and if it
does the customer received a service at
half price
one percent of its customers received a
service for half price so we're gonna
build our model to describe oil change
times so the random variable X
represents the time to complete the oil
change we have a mean time of 19 minutes
and where are my inflection points gonna
occur at 17 one standard deviation below
the mean 21 one standard deviation above
the mean in this particular problem we
want to know the proportion of oil
changes that are gonna last longer than
20 minutes cuz we want to know what
proportion of customers are gonna get
half off so the area of that shaded
region is what we're looking for yes so
we can do this by hand by converting to
a z-score and then we would go to our
table and find the area left of that
z-score and to find this area we would
subtract that result from from one
instead what we're gonna do is go into
statcrunch she is oh wait I got my
calculator right there so the mean was
19 the standard deviation was 2 and we
want to know the probability greater
than 20 that's the same as the
proportion greater than 20 hit compute
and we get our result
so my answer is point 3 0 8 5 all they
wanted is a percent so that is thirty
point eight five percent and we could
enter that in here or B if the
Automotive Center does not want to give
a discount to more than three percent of
its customers how long should it make
the guaranteed time limit so here in
Part B again we can use our normal model
to help us answer the question well
that's a really bad one let's see the
mean is 19 and we want to only give this
discount to three percent of the
customers so what do you think that
means
well you point where the point A three
go I want you to I want I want to know
if my mater is gonna look like now let
me ask you this if I set my time here do
you think that that would mean I give my
discount to three percent of the
customers
well what percent of my customers are
gonna have an oil change of 19 minutes
or less ask a different question what's
the area under the normal curve to the
left of the mean point five fifty
percent this way fifty percent that way
so if I establish my cutoff here and
give discounts to any time bigger than
whatever this acts is am I gonna be
helped am I going to be giving discounts
to less than 50 percent or more than 50
percent
well if my old change is bigger than axe
they get a discount correct so it's
going to be 50% of the eat there plus
this area so it's gonna be well over 50%
are gonna get a discount okay so do you
think that my value for cutting for oil
change time should be this value no I
would lose my shirt right so again with
that idea in mind I want to give a
discount I don't want to but I'm going
to give a discount to 3% of my customers
based on an oil change that takes too
long cuz you're ticked off that you're
in the waiting room and you've got
things to do so I'm gonna start here and
you tell me where you think I should
stop to mark for my model for 3% get a
discount meaning 97% don't here we go
stop here around here is going to be my
cutoff unknown X and then I'll shade
this area here like should that area be
a three meaning this over here is 0.97
those folks in this region Noorie
discount over here just count so we want
to find the value of x such that the
area to the right is point zero three if
we're doing this by hand we have to go
into our table look for an area closest
to point nine seven find the Z value
that corresponds to an earlier point
nine seven and then use this x equals mu
plus z sigma formula right if we're
going to do this in statcrunch we enter
or
I mean and our standard deviation in
there and we could do this one a to race
if I type in point nine seven I would
want probability less than and I get
twenty two point seven six see that
although I could type in 0.03 over here
and do probability greater than and I
get the same thing twenty two point
seven six either way I'm going to
establish as my cutoff or for you all
changes at twenty two point seven six
minutes which is kind of a bizarre value
but it sounds cool right it's a
marketing ploy of sorts all right if you
think about it if your oil change is not
gone in twenty two minutes and forty
five point three seconds you get a 50%
discount right now when you say you'd be
sitting there in the waiting check and
watching your watch wouldn't you where
she would I like this ploy let's make
sure we're right what did I say twenty
round to the nearest integer they want
twenty three minutes as a boring I like
our twenty two point seven six minutes
how did I get to twenty two point seven
six okay this area right here is point
zero three okay so what we're doing here
is looking for the cut off the x value
such that the area to the right is point
zero three or another way to think about
it the area to the left is point nine
seven meaning ninety seven percent of
the folks getting an oil change and not
going to get a discount than three
percent whoa
yes what you would have to do we might
as well do this for fun so yeah if we're
doing this by hand I got to go to my
normal table sorry for the size here in
this tea where we look for an area
closest to 0.97 okay so point nine seven
point nine three I got point nine six
nine nine right door and that
corresponds to a Z value of one point
eight eight see it's in the 1.8 row okay
and the second the last column I think
is point oh eight yes so the Z value is
one point eight eight so I would say
that x equals mu is 19 plus Z is one
point eight eight times Sigma is 2 all
right what does that I work out too hey
same thing