Probabilities with the Sample Mean Video Lecture | Statcrunch Tutorials for Beginners - Data & Analytics

hi again we're going to do a problem
from section 6.5 the central limit
theorem section we're going to number 11
on page 309 so I'll read the problem
says a ski gondola and Vail Colorado
Carrie skiers to the top of the mountain
it bears a plaque stating that the
maximum capacity is 12 people or two
thousand four pounds that capacity will
be exceeded if 12 people have weights
with the me and greater than 2000 4/12
which is 167 because men tend to weigh
more than limit a worst case scenario
involves 12 passengers who were all men
men have weights that are normally
distributed with a mean of 172 and
standard deviation of 29 pounds so once
the probability will do part be here
find the probability that 12 ran was
selected men will have a mean that is
greater than 1 67 okay so if X is the
weight of a randomly selected male and
we know from the problem that X is
distributed as normal random variable
with mean 172 standard deviation of 29
what we're trying to find is the
probability that that total weight is
greater than 2004 okay so this is the
same it's the probability if I just go
ahead and divide each of these by 12 the
total divided by 12 is just the average
x-bar 12 and we want to find the part
that the probability of X bar 12 is
greater than 2000 4/12
and 2004 / 12 is that 167 I'm just going
to write 167 here to save little space
okay well what do we know about X bar 12
it's not normal 172 29 x bar 12 we know
is normal with mean 172 same as the
original random variable but the
standard deviation is not 29 its twenty
nine divided by the square root of 12
okay and we've got an approximation for
that it's 8.3 716 so this is the
approximate standard deviation of X bar
12 okay so now we just want to find the
probability that X bar 12 is greater
than or equal to 1 67 and to do that we
would standardize so the probability of
X bar 12 we'll just go ahead and
subtract the mean which is call me call
you x bar going right the number in and
we'll divide by sigma x bar and here
will standardize so mu x bar is equal to
1 72 and Sigma X bar is equal not equal
to 29 is equal to the 8.37 16
so and since i took a normal random
variable and standardized it this is see
and hear will take find the z-score take
160 7-1 72 and we'll divide that by 8.3
716 so it's negative point 5973 say 5973
so if you wanted to look this up in your
table it would be all the area to the
right of negative point six so 7257 so
that's the chances that if you just
randomly chose 12 min that they would be
more than that 2004 if you wanted to use
statcrunch from them
again the way to get to statcrunch is go
ahead just use the link for the moon
alone page alright so we again want to
use that this is a normal random
variable so we want to use stat and go
down to calculator and then normal if
you wanted to just use the if you wanted
to not be have to use the table in the
back your book but to get something very
similar you would just use a z-table so
normal 0 1 and you want to know how much
area is there to the right of negative
16 this is this is exactly what we
looked up in the table and it's 7257
just like we saw before if you and I
believe this is the way that mystatlab
wants you to answer it if you wanted to
do this a little bit more efficiently it
may be a little more accurately what you
would do is say that okay a random
variable has a mean of 172 and has a
standard deviation of 8.3 716 it's best
to round up to four places by the way
and we want to find the probability that
that is bigger than 167 notice there's
not an option it's just greater than or
equal to there's not an option for dis
greater because with normal random
variables that's exactly the same so I
did compute and I get 7240 so 7257 7248
almost exactly the same this is a little
more accurate but you just have to I'm
not sure exactly the way that mystatlab
wants that so that will do it for this
one just remember when you're doing
probabilities with X bar the mean is
going to be the same when you
standardize it but you / not Sigma u
divided by Sigma divided by the square
root of n so that's it for today