Sampling distribution of the difference in sample means

WEBVTT
Kind: captions
Language: en

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- [Teacher] What we're
going to do in this video

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is explore the sampling
distribution for a difference

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in sample means, and we'll use
this example right over here.

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So it tells us a large bakery

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makes thousands of cupcakes
daily in two shifts:

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shift A and shift B.

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Suppose that, on average,
cupcakes from shift A

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weigh 130 grams with a
standard deviation of 4 grams.

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For shift B, the mean
and standard deviation

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are 125 grams and 3 grams, respectively.

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Assume independence between shifts.

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Every day, the bakery takes
a simple random sample

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of 40 cupcakes from each shift.

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They calculate the mean
weight for each sample,

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then look at the difference, A minus B,

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between the sample means.

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Find the probability that the
mean weights from the samples

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are more than 6 grams
apart from each other.

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So I'm actually not gonna
tell you immediately

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to pause this video

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and try to work through this on your own.

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First I'm gonna think about
how we could break this down,

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and then I'll ask you to pause

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and try to tackle each
of those parts by itself.

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So in order to tackle
this eventual question,

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we're going to have to
think about the mean

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of the sampling distribution

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for the difference in sample means.

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So sample mean from group A
minus sample mean for group B.

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We're gonna have to think
about the standard deviation

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of the sampling distribution

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for the difference in sample means.

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And we're going to have to think

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about is this distribution normal?

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If we're able to figure
out these three things,

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then we just have to figure out,

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well, how many standard
deviations away from the mean

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is this?

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And we could use your standard z-table

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to figure out the probability.

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So now I encourage you to pause this video

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and try to tackle this first part.

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What is the mean of the
sampling distribution

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for the difference in sample means?

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All right, now let's work
through this together.

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So the mean of the sampling distribution

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for the difference in sample means,

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and we have seen this before,

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this is going to be
equal to the difference

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between the means of the
sampling distribution

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for each of the sample means.

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So that mean minus this mean.

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And we also know that the mean
of the sampling distribution

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for each of these sample means,

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that's just going to be
the mean of the population

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that we are sampling from.

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So this mean right over here

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is just going to be the
mean, the population mean,

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for shift A, which is gonna be 130 grams.

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I'll just write that there.

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And then the mean of the
sampling distribution

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for the sample means from shift B,

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we can see that that's just
going to be the population mean

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for shift B, which is right over here.

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So minus 125 grams.

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And of course, this is just
going to be equal to 5 grams.

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So we have answered the first part.

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We know the mean of the
sampling distribution

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of the difference in sample means.

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Now what about the standard deviation?

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So for that, let's think
actually about variances

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'cause the math's a little
bit easier with variances.

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And then from that, we can
derive standard deviations.

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So we know that the variance
of the sampling distribution

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for the difference in sample means,

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assuming that your two
samples are independent

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and you're sampling with replacement,

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if you're sampling with replacement,

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it's actually going to be
the sum of the variances

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of the sampling distribution
for each of the sample means.

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So it's going to be that
plus this right over here.

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Now you might be saying,

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"Wait, we're not sampling
with replacement."

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Well, we also know that if
each of the sample sizes

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are less than 10% of the population,

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then the difference is negligible,

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and so we could still use this formula.

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And so you could see that
the simple random sample here

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is 40 from each shift.

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And they say that a large bakery

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makes thousands of cupcakes
daily in two shifts.

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So even if it was a thousand,
10% of that would be 100,

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this is less than 10%.

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So we meet that condition, so
we can use this same formula

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that you would use if you were
sampling with replacement.

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So this first variance right over here

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of the sampling distribution

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for the sample means from shift A,

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this is going to be equal
to the variance of shift A,

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the population variance of shift A

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divided by your sample size.

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And then this over here,

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it's gonna be the same thing for shift B.

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It's going to be the variance of shift B

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divided by your sample size.

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And so this is going to be equal to what?

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Well, the variance from shift A

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is going to be the square of
the the standard deviation

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from shift A.

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The standard deviation's right over there.

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And so that's going to be 16.

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We could write grams squared

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if we wanna keep the units there.

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And then we're going to
divide by the sample size.

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We know that the sample size in each case,

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40 cupcakes at a time for each sample.

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And then for shift B, we know
that the standard deviation,

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the population standard
deviation for shift B is 3 grams.

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You square that, and
you get 9 grams squared.

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A gram squared is kind
of an interesting idea,

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but that's what the units are
working out to be right now.

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And our sample size is still equal to 40.

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And so this is going to be equal to,

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let's see, 16 plus 9 is 25.

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Common denominator, 40.

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So it's 25 over 40, which
is the same thing as 5/8,

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5/8 of a gram squared, which
is a little bit strange unit,

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but this now tells us

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what the standard deviation is going to be

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because it's going to be the square root

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of all of this business.

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So the standard deviation
of the sampling distribution

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for the difference in
sample means over here

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is going to be the square root of 5/8.

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And now of course, the
units are back to grams,

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which makes sense.

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And this is approximately
going to be equal to,

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get my calculator out,
5 divided by 8 equals,

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and then we take the square root of that,

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and it's going to be approximately 0.79.

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0.79.

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So the next question,

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before we try to figure
out the probability is,

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is are we dealing with a
normal distribution here

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when we think about the
sampling distribution

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for the difference in sample means?

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And so I encourage you
to pause the video again

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and think about that.

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So there's two ways that we can assume

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that the sampling distribution
for the difference

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in sampling means is normal.

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If the original populations
that each of the sample means

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are being calculated from are normal,

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then that means that the
sampling distribution

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for each of the sample
means is gonna be normal.

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And that means that the difference

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of the sampling distributions
are going to be normal.

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Now we don't know for a fact

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that the weights of the cupcakes

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from each shift are normal distributions,

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but we also know that
the sampling distribution

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of the sampling means

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can be modeled as being
approximately normal

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if the two sample sizes are
greater than or equal to 30.

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And we know that each of these samples

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are definitely greater than
or equal to 30, they are 40.

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So that tells us that
the sampling distribution

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of the difference in sample
means is also normal.

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So we've established
the things that we need

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to then calculate the probability.

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So I encourage you, pause the video,

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and see if you can use that information

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to calculate that probability,

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and we will then do
that in the next video.