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Statistics – Sampling Distribution – Sample less than 5% w/TI-83/84 Calculator

October 21, 2019

This is Chris Ferbrache and I’m
going to perform a calculation using a sampling distribution and we’re actually
going to find the probability of a certain value occurring or greater than
or equal to type of value okay population mean a population mean who
received money from the IRS is $1100 for the standard deviation of $500 so this
our population mean is 1100 and 500 for our standard deviation if a random
sample of 50 individuals receiving a tax return are selected what’s the
probability the sample mean should be sample mean ia’s will be more than
$1,200 let’s write out let’s write out what we know right off the bat and what
we know is we have a Mew of $1100 a population standard deviation of $500
and we’re looking for the probability of our sample mean being $1,200 and our
sample size this lowercase n is 50 it’s a couple of things here right off the
bat you may notice we do not have a sample standard deviation and what we’re
gonna do is we’re gonna calculate the standard error of X or the standard
deviation of the sampling mean and we do that by by using this formula okay so let’s write down our probability
or our statement of what we’re we’re actually trying to solve for so we’re
looking for X bar is more than 1200 make sure your inequalities pointing the
correct direction it’s very important because it’s going to help us make sure
that we’re right when we do the calculation because it goes and points
towards our area of interest in our area of interest is right here in this tail it’s our area of interest right here on
the right bigger I always write the MU in here this kind of gives us the
perspective of the whole problem so our mu is 11 okay so we’re doing good here
okay let’s solve this a couple things are right off the bat we have a
population we have a population standard deviation and a mean and we have a
sample that’s greater than 30 so the normal thing that I would do here
is say are we going to use the t-distribution are we going to use a Z
distribution but we have a decent sized sample and we have a population standard
deviation and and population mean we can just do the Z distribution we don’t want
to use any other one for this so we’re gonna go and use our Z distribution so we want to find a z-score so that we
can have it all standardized and figure out what the probability is so we’re
looking for the z-score and it’s going to be x mean this is the point estimate
that we’re looking for that’s remember 1,200 from 1100 and then this part right
here let’s calculate that out it’s going to be what is that 500 this is our
standard error of X that were calculating so 500 over the root the
square root of n which is 50 and our answer I have it worked out it is 70
point and seven one zero six so so that’s a that’ll work fine okay
so down below here we have we have 70 point seven one zero six and when we
worked this out for our Z value with all of these numbers plugged in we end up
with an answer of 1.4142 and what we want to do with that is we
would go and look this up on the Z distribution table if we look this up on
the Z table we end up coming back with the value under the value under the
curve beam Oh point four two zero seven so we have
point four two zero seven this is the area under the curve of one side so we
do this the sides point five and then we know that this side over here is 0.42
zero seven so if we go we work this out we have to go and there’s a couple ways
to go about this we can go and subtract this value that we got from underneath
the curve from 0.5 and we’ll get the answer or we can add both of these up
subtract it from one doing the same thing we’re just including the point
five we have to count for that whether we subtract from 0.5 or subtract from
one so let’s do this the common way to do it I’d say would be take point five
plus point four two zero seven it’s probably a little bit more logical and
we end up with 0.92 zero seven so that’s actually the answer if we’re trying to
solve for what it would be less than 1200 so you see we have this whole area
we just solve for x this would be less than 1200 but we’re solving for more so
we want to subtract it from 1 9 to 0 7 and we end up with point zero seven nine
three and writing sometimes with the marker it’s a little tricky point zero
seven nine three and this would be our answer for this problem this equates to
this right there so let’s do this on the TI 83 84 variant I have an emulator
ready to go I will pull up so the emulator just get it on the
corner here okay let’s turn this on so to do this we go into the we want to go
into the sampling distribution area for this and so we go into the normal CDF
and whether it’s normal whether it’s a sampling distribution or not a sampling
distribution we use the normal CDF which is the cumulative density function and
the only real difference is that we because it’s a sampling distribution
we’re using essentially the standard error of X as our standard deviation
right here so we don’t have the we’re making an estimate on what our standard
deviation is so we have to use this as our standard deviation which is fine for
what we’re working with so we’re gonna do seconds no I’m not okay second D is
TR and we end up with a bunch of different distribution programs within
the calculator so we’re gonna do normal CDF because we’re looking for a
cumulative value we’re looking for a greater than in this case with
cumulative distributions the difference between equals 1200 and greater or
greater than so whether it’s greater than 1200 or equals 1200 and greater the
difference is so my new for us it doesn’t really matter it’s cumulative so
it’s it’s a tiny tiny difference not enough to worry about so we have our
normal CDF selected enter okay this is the thing that always comes up people
say well what should I do about this so for the lower bound in this case our
lower bound is going to be 1200 that that makes sense right
because we’re sure we’re solving for 1200 and up our upper our upper if in
doubt there’s couple things we can do for our upper as a minimum we should do
this it would be mu the value of meal plus the standard deviation times 4
right that it’d be mu plus 4 times what our estimated standard deviation is or
our error –air of X so what we’re doing is we’re taking mu say our mu is 1100
and then we are multiplying what we calculated as your standard deviation or
air of X in this case multiply it by 4 that puts us so far out in the positive
side that we’re not cutting off some of the probability values that could occur
remember three standard deviations is 99.7 four I’m not sure but it’s it’s
very high it’s of course higher than that so the other thing that we can do
is we do this thing that’s in the calculator that’s the highest number
that the calculator will do and it’s 1 and then second e and you do 99 and
that’s really really the biggest number they can do so it’s fine we’re not gonna
lose any precision by doing this we’re gonna gain precision okay now our mu is
1100 and our standard deviation is the standard deviation that we calculated
right here so make sure you don’t put it in the population standard deviation
you’ll have to put in this value right here otherwise you’re not going to be
it’s not gonna be right so it’s just a couple things we can do we could
actually put in the formula right here in there which is really kind of neat
but we already have it calculated so put in 70 point seven one zero six and it just Prime’s it up if you’re
using an older calculator you might not have that nice little wizard interface
then you have to put in the values in in the parenthesis with commas okay let’s
see if our numbers match up now that’s the interesting thing here that’s in the
number so it’s 0.0 0.6 and I point zero seven nine three so we’re pretty close
but I’ll have to go back and do a little bit of checking on this it’s possible I
should have rounded up two when I when I picked out the I don’t have the Z table
right in front of me but when I picked out the Z table value it’s possible that
I should have went up to the next value on the Z table and then I would have
come up with a slightly different calculation here but as you see I’m
pretty close it’s not it’s not the most comforting feeling that that I’m a
little off on this with it being by the third digits off by one it sounds like
it’s not a lot but this would make me scratch my head and go back and check so
that might be something that I’ll do other or not everything’s good we did
the math correct the calculator is always more accurate than than we are
because when we use the tables they don’t go out as accurate as the
calculator okay well I hope you learned a little bit about standard about
sampling means with continuous probability distributions.

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