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Finding area under the normal curve using StatCrunch Video Lecture - Statcrunch
Text Transcript from Video
steel rods are manufactured with the
mean length of 21 centimeters because of
variability in the manufacturing process
the lengths of the rods are
approximately normally distributed with
a standard deviation of point zero five
centimeters so that means that if we
want to draw our model there's our
normal curve where X represents the
length of a randomly selected rod the
mean is going to be 21 and then we're
gonna have inflection points here and
here
what would this value be well this one
would be twenty point nine five point
zero five standard deviation I'm sorry
point zero five below the mean right one
standard deviation below the mean and
then this value is going to be point
1.05 one standard deviation above the
mean so we want to know the proportion
of steel rods less than twenty point
nine and so if I label twenty point nine
on my model the proportion left would be
represented by this area so if we agree
that we're going to use statcrunch to
find that area stat calculators and we
want the normal calculator our mean was
what
more than our standard deviation Barb
and we want the area left of 20.9 it's
like a compute and there's our result
grabbed us copy it and drop it in here
and so our answer right there would be
it'll be good point o2 to eat Oh two to
eight fantastic I have any rod shorter
any write shorter than twenty point
eighty or a longer than twenty one point
one two now when we draw a picture of
this scenario I want you to observe
something that's kind of interesting
there was our normal model mean at
twenty one twenty point 88 I've just
generically throw in there let's say
it's right around there well that would
be the number rot proportionate rod
shorter than twenty point eight eight
right and over here is what 21.1 to do
you notice that that 20 point AE is
exactly point one two units below the
mean see that and the 21 point one two
is 0.12 units above the mean and so
because of the symmetry under the normal
code what could we say about this area
in relation to that area they're going
to be the same area so rather than
spending time finding this area and then
separately finding this area what we
could do to
efficient is simplifying this area and
then add a tweet itself or double the
value okay so that's what we'll do
that's more of an issue when you're
doing this by hand than it is if you're
using something like statcrunch because
twenty point eight eight right and now
less than we get an area at point zero
zero eight to see that oh and you're
going to do this online okay to verify
that what i said is accurate if i type
in any one point one two I get the same
value point oh eight two so the area
right there there's gonna be two times
that point
0:08 two and so we get what one six four
using the results of Part B of five
thousand rods are manufactured in a
daily how many should the plant manager
expect to discard so now we sort of have
a binomial situation going on because
either we're going to get a success
meaning we don't meaningly discard or we
gotta fill in your meaning we don't
discard those successes finding of a rod
that we're going to discard meaning
finding a rod whose length is shorter
than twenty point eight eight
centimeters or longer than twenty-one
point one centimeters now the
probability of finding a rod that gets
discarded is what
well the probability that this started
rod is point oh one six four and so the
expected number of rods to be discarded
is going to be yeah we're going to use
the formula for the mean because the
mean of the random variable X is the
same as its expected value right and
here since it's a binomial the expected
value is n times P so we have 5,000 rods
times point zero one six four and that's
going to give us on the nose 82 really
nice 82 and lastly if an order comes in
for 10,000 steel rods how many should
the plant manager expect to manufacture
if the order states that all rods but
must be between twenty point nine and
twenty one point one centimeters first
off we've got to figure out the
proportion that would be discarded okay
and so if we go back to Part A the
proportion of rods less the twenty point
nine centimeters is point zero two to
eight right so what do you think the
proportion of rods greater than twenty
point one centimeters would be
if I draw a picture of this scenario
there's 21 right 20.9 we know is right
there something like that right and we
get the a 0.022 eight twenty point one
it's over here twenty one point one is
over here it's point one units above the
mean
this is 0.1 units below so what do you
think that area is also 0.02 to eight
because of symmetry because its symmetry
so what proportion of rods do I expect
to discard all I expect to discard any
less than point twenty point nine I'm
going to discard any that are greater
than twenty-one point one so the
proportion that are going to be
discovered it is yeah point oh four five
six is proportion discarded just add up
these two proportions okay Wow that's
how many I'm going to get rid of how
many am I going to keep or what
proportion I going to keep I should say
well - that zero point nine five four
six all right and so that's the poor
alright nine five four four making sure
you're paying attention that's the
proportion kept right now overall how
many do I want to be able to deliver
according to the problem how many steel
rods don't want to be able to deliver
ten thousand so I want to manufacture a
certain amount of rods so that the
expected number of rods I'll be able to
keep is ten thousand so it's a little
different than up here but we use the
same formula that the expected value of
X is n times P well in this problem I
want my expected number of rods tapped
to be ten thousand I don't know how many
I need to manufacture that I know the
proportion that I'm going to keep is
0.95 for four so to get and I just
compute ten thousand divided by 0.9 five
four four and that gives me say that
again one point four seven one four
seven eight on the nose rounded which
okay
run up okay so we're going to
manufacture ten thousand four hundred
seventy eight we expect that we'll have
to discard 478 so that will ultimately
be able to deliver 10,000 steel rocks
let's see if that pans out according to
program 10,000 478 excellent
mean length of 21 centimeters because of
variability in the manufacturing process
the lengths of the rods are
approximately normally distributed with
a standard deviation of point zero five
centimeters so that means that if we
want to draw our model there's our
normal curve where X represents the
length of a randomly selected rod the
mean is going to be 21 and then we're
gonna have inflection points here and
here
what would this value be well this one
would be twenty point nine five point
zero five standard deviation I'm sorry
point zero five below the mean right one
standard deviation below the mean and
then this value is going to be point
1.05 one standard deviation above the
mean so we want to know the proportion
of steel rods less than twenty point
nine and so if I label twenty point nine
on my model the proportion left would be
represented by this area so if we agree
that we're going to use statcrunch to
find that area stat calculators and we
want the normal calculator our mean was
what
more than our standard deviation Barb
and we want the area left of 20.9 it's
like a compute and there's our result
grabbed us copy it and drop it in here
and so our answer right there would be
it'll be good point o2 to eat Oh two to
eight fantastic I have any rod shorter
any write shorter than twenty point
eighty or a longer than twenty one point
one two now when we draw a picture of
this scenario I want you to observe
something that's kind of interesting
there was our normal model mean at
twenty one twenty point 88 I've just
generically throw in there let's say
it's right around there well that would
be the number rot proportionate rod
shorter than twenty point eight eight
right and over here is what 21.1 to do
you notice that that 20 point AE is
exactly point one two units below the
mean see that and the 21 point one two
is 0.12 units above the mean and so
because of the symmetry under the normal
code what could we say about this area
in relation to that area they're going
to be the same area so rather than
spending time finding this area and then
separately finding this area what we
could do to
efficient is simplifying this area and
then add a tweet itself or double the
value okay so that's what we'll do
that's more of an issue when you're
doing this by hand than it is if you're
using something like statcrunch because
twenty point eight eight right and now
less than we get an area at point zero
zero eight to see that oh and you're
going to do this online okay to verify
that what i said is accurate if i type
in any one point one two I get the same
value point oh eight two so the area
right there there's gonna be two times
that point
0:08 two and so we get what one six four
using the results of Part B of five
thousand rods are manufactured in a
daily how many should the plant manager
expect to discard so now we sort of have
a binomial situation going on because
either we're going to get a success
meaning we don't meaningly discard or we
gotta fill in your meaning we don't
discard those successes finding of a rod
that we're going to discard meaning
finding a rod whose length is shorter
than twenty point eight eight
centimeters or longer than twenty-one
point one centimeters now the
probability of finding a rod that gets
discarded is what
well the probability that this started
rod is point oh one six four and so the
expected number of rods to be discarded
is going to be yeah we're going to use
the formula for the mean because the
mean of the random variable X is the
same as its expected value right and
here since it's a binomial the expected
value is n times P so we have 5,000 rods
times point zero one six four and that's
going to give us on the nose 82 really
nice 82 and lastly if an order comes in
for 10,000 steel rods how many should
the plant manager expect to manufacture
if the order states that all rods but
must be between twenty point nine and
twenty one point one centimeters first
off we've got to figure out the
proportion that would be discarded okay
and so if we go back to Part A the
proportion of rods less the twenty point
nine centimeters is point zero two to
eight right so what do you think the
proportion of rods greater than twenty
point one centimeters would be
if I draw a picture of this scenario
there's 21 right 20.9 we know is right
there something like that right and we
get the a 0.022 eight twenty point one
it's over here twenty one point one is
over here it's point one units above the
mean
this is 0.1 units below so what do you
think that area is also 0.02 to eight
because of symmetry because its symmetry
so what proportion of rods do I expect
to discard all I expect to discard any
less than point twenty point nine I'm
going to discard any that are greater
than twenty-one point one so the
proportion that are going to be
discovered it is yeah point oh four five
six is proportion discarded just add up
these two proportions okay Wow that's
how many I'm going to get rid of how
many am I going to keep or what
proportion I going to keep I should say
well - that zero point nine five four
six all right and so that's the poor
alright nine five four four making sure
you're paying attention that's the
proportion kept right now overall how
many do I want to be able to deliver
according to the problem how many steel
rods don't want to be able to deliver
ten thousand so I want to manufacture a
certain amount of rods so that the
expected number of rods I'll be able to
keep is ten thousand so it's a little
different than up here but we use the
same formula that the expected value of
X is n times P well in this problem I
want my expected number of rods tapped
to be ten thousand I don't know how many
I need to manufacture that I know the
proportion that I'm going to keep is
0.95 for four so to get and I just
compute ten thousand divided by 0.9 five
four four and that gives me say that
again one point four seven one four
seven eight on the nose rounded which
okay
run up okay so we're going to
manufacture ten thousand four hundred
seventy eight we expect that we'll have
to discard 478 so that will ultimately
be able to deliver 10,000 steel rocks
let's see if that pans out according to
program 10,000 478 excellent
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Apr 22, 2026 Last updated
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