When is the normal approximation considered reasonable?

The normal approximation is considered reasonable if both n*p and n*(1-p) are greater than or equal to 10.

What is continuity correction and why is it used?

Continuity correction is used to adjust for the differences between discrete and continuous distributions, improving the accuracy of normal approximations to binomial probabilities.

How does one perform standardization in the context of this approximation?

Standardization is done using the formula Z = (X - μ) / σ, where X is the binomial random variable, μ is the mean, and σ is the standard deviation.

What does a continuity correction imply when calculating probabilities?

A continuity correction implies adjusting the value at which you calculate probabilities to better represent discrete values in a continuous setting, such as using 51.5 or 52.5 instead of 52.

How does sample size affect the normal approximation's accuracy?

As the sample size increases, the approximation to the normal distribution improves, making it a better fit for the binomial distribution.

What are the conditions under which the binomial distribution is perfectly symmetric?

The binomial distribution is perfectly symmetric when the probability of success (p) is exactly 0.5.

How can one find the exact probability for binomial distributions?

Exact probabilities for binomial distributions can be calculated using the binomial formula or computational tools.

Can the success probability p affect the shape of the binomial distribution?

Yes, the binomial distribution becomes skewed when p is not equal to 0.5, especially as it approaches 0 or 1.

What is an example of applying the normal approximation?

An example includes calculating the probability that X is greater than or equal to 52 using the normal approximation with continuity correction.

The Normal Approximation to the Binomial Distribution

00:14:10

https://www.youtube.com/watch?v=CCqWkJ_pqNU

Summary

TLDRThe video discusses the normal approximation to the binomial distribution, highlighting its utility in simplifying calculations historically and its relevance today in statistical inference. It explains that the approximation is most accurate when the success probability is around 0.5 and the sample size is large. The video demonstrates the application of continuity correction, where adjustments to the standardization process significantly enhance the accuracy of probability estimates. Examples illustrate the difference in results with and without this correction, underscoring its importance in practical applications.

Takeaways

📈 The normal approximation simplifies binomial calculations.
🔍 Approximation works best with p close to 0.5 and larger n.
🔧 Continuity correction improves accuracy when transitioning from discrete to continuous distributions.
📊 The binomial distribution is symmetric when p = 0.5.
📝 Standardization involves converting binomial variables to Z-scores.
🔑 Large sample sizes enhance the normal approximation's fit.
📊 n*p and n*(1-p) should both be >= 10 for a reasonable approximation.
🔄 Exact binomial probabilities can still be computed for accuracy.
⚠️ Understanding the logic behind calculations aids in applying them correctly.
🏷️ The shape of the binomial distribution varies with p values.

Timeline

00:00:00 - 00:05:00
The video discusses the normal approximation of the binomial distribution, explaining the advantages of this approximation in simplifying probability calculations. The discussion notes that the binomial distribution is symmetric at p = 0.5 and becomes skewed for values of p away from this point. Using illustrations of distributions, the speaker clarifies that as sample size increases, the normal approximation becomes more accurate, particularly when n*p and n*(1-p) are both greater than or equal to 10. The importance of understanding this guideline for normal approximation in statistical analysis is emphasized.
00:05:00 - 00:14:10
The latter part of the video focuses on the use of continuity correction to improve normal approximation. It explains how adjustments are made when transitioning from a discrete binomial distribution to a continuous normal distribution, highlighting the necessity of starting calculations at values like 51.5 or 52.5 to accurately include or exclude specific outcomes. Through various examples and probability calculations using continuity corrections, the speaker demonstrates how these adjustments lead to significantly closer approximations to exact binomial outcomes. This approach enhances the overall accuracy of statistical estimates.

Mind Map

Video Q&A

What is the normal approximation to the binomial distribution?
The normal approximation is a technique to estimate probabilities in a binomial distribution using the continuous normal distribution, particularly useful when dealing with larger sample sizes.
When is the normal approximation considered reasonable?
The normal approximation is considered reasonable if both n*p and n*(1-p) are greater than or equal to 10.
What is continuity correction and why is it used?
Continuity correction is used to adjust for the differences between discrete and continuous distributions, improving the accuracy of normal approximations to binomial probabilities.
How does one perform standardization in the context of this approximation?
Standardization is done using the formula Z = (X - μ) / σ, where X is the binomial random variable, μ is the mean, and σ is the standard deviation.
What does a continuity correction imply when calculating probabilities?
A continuity correction implies adjusting the value at which you calculate probabilities to better represent discrete values in a continuous setting, such as using 51.5 or 52.5 instead of 52.
How does sample size affect the normal approximation's accuracy?
As the sample size increases, the approximation to the normal distribution improves, making it a better fit for the binomial distribution.
What are the conditions under which the binomial distribution is perfectly symmetric?
The binomial distribution is perfectly symmetric when the probability of success (p) is exactly 0.5.
How can one find the exact probability for binomial distributions?
Exact probabilities for binomial distributions can be calculated using the binomial formula or computational tools.
Can the success probability p affect the shape of the binomial distribution?
Yes, the binomial distribution becomes skewed when p is not equal to 0.5, especially as it approaches 0 or 1.
What is an example of applying the normal approximation?
An example includes calculating the probability that X is greater than or equal to 52 using the normal approximation with continuity correction.

View more video summaries

Get instant access to free YouTube video summaries powered by AI!

Subtitles

Auto Scroll:

00:00:02
let's take a look at the normal
00:00:03
approximation to the binomial
00:00:05
distribution The Continuous normal
00:00:07
distribution can sometimes be used to
00:00:09
approximate the discrete binomial
00:00:13
distribution why would we want to do
00:00:15
this in the olden days it was very
00:00:16
useful for probability calculations the
00:00:18
binomial formula can be a bit of a pain
00:00:20
if it has to be used over and over and
00:00:22
over again and so this normal
00:00:24
approximation came in to make lives much
00:00:26
easier these days the computers can do
00:00:28
the calculations for us so it's not as
00:00:29
much of an issue there but we still use
00:00:33
this normal approximation in statistical
00:00:35
inference when we do things a little bit
00:00:36
later on like statistical inference for
00:00:38
proportions we often use this normal
00:00:40
approximation so it's good to know a
00:00:42
little bit about
00:00:43
it now you may recall that the binomial
00:00:46
distribution is perfectly symmetric if p
00:00:50
is exactly equal to 0.5 and will have
00:00:52
some skewness when p is not equal to 0.5
00:00:56
now the normal distribution is a
00:00:58
symmetric distribution and so the normal
00:01:00
approximation is going to work best when
00:01:02
p is close to 0.5 and it's going to work
00:01:04
better and better as we get a larger and
00:01:06
larger sample size as
00:01:08
well for illustrative purposes here is
00:01:11
the binomial distribution with n is 40
00:01:13
and P is .5 and as you may be able to
00:01:16
tell this is a perfectly symmetric
00:01:18
distribution in this case and let's say
00:01:20
we were to superimpose a normal curve
00:01:23
over this
00:01:24
distribution it looks awfully normal
00:01:26
that superimposed normal curve fits
00:01:28
pretty well and we'd see that if we
00:01:30
jacked up the sample size even higher
00:01:32
and higher and higher that normal curve
00:01:35
would fit better and
00:01:37
better now what if we're a little bit
00:01:39
closer to the boundary here here p is 03
00:01:43
which is close to the boundary of zero
00:01:45
and we've got the same value of n but we
00:01:47
see a little bit of skewness and if we
00:01:49
were to superimpose a normal curve it
00:01:51
does not fit very well if we amped up
00:01:55
the sample size larger and larger we'd
00:01:57
see that the normal approximation is
00:01:59
going to be better and better as that
00:02:01
sample size gets larger and larger but
00:02:03
the general idea here when we get near
00:02:05
the boundaries of zero or one we are
00:02:08
going to need a larger and larger value
00:02:10
of n for that normal approximation to be
00:02:14
reasonable and this idea summarized in
00:02:16
this rough guideline that the normal
00:02:18
approximation is reasonable if both n *
00:02:21
p is bigger than or equal to 10 and n *
00:02:24
1 minus p is bigger than or equal to 10
00:02:26
and if you play around with that a
00:02:27
little bit you'll see simply that if p
00:02:29
is close to 0 or 1 we need a larger
00:02:32
value of n in order for the normal
00:02:33
approximation to be reasonable now this
00:02:36
is just a rough guideline sometimes
00:02:38
people replace 10 with five here and
00:02:41
sometimes use different rules Al
00:02:43
together so you should consult with your
00:02:44
professor or your textbook to see what
00:02:46
rough guideline they are
00:02:48
using recall that if x is a binomial
00:02:51
random variable then X has a mean of n *
00:02:54
p and a variance of n * P * 1us p and
00:02:59
what we were just disc discussing above
00:03:00
is that X can be considered
00:03:02
approximately normal in certain settings
00:03:04
and so then we can standardize this in
00:03:07
the usual way we can say
00:03:09
xus mu over Sigma using mu from up here
00:03:15
and sigma being the square root of Sigma
00:03:17
squar from here then this quantity we're
00:03:21
going to call a zed and we're going to
00:03:23
call that Zed because that quantity has
00:03:26
approximately the standard normal
00:03:28
distribution so my Zed is going to be
00:03:32
approximately standard normal normal
00:03:34
with a mean of zero and a variance of
00:03:36
one and now we're going to use this in
00:03:38
probability
00:03:39
calculations so let X be a binomial
00:03:42
random variable with n of 75 and P equal
00:03:45
to6 what is Mu well we know mu is equal
00:03:49
to n * p and that's going to be 75 * 0.6
00:03:53
and that works out to 45 what is Sigma
00:03:56
squar n * P * 1 - p and that is going to
00:04:02
be equal to 75 * 0.6 * 1 -
00:04:08
0.6 and that works out to 18 and so our
00:04:11
standard deviation is simply going to be
00:04:13
the square root of
00:04:15
18 now suppose we wanted to use the
00:04:17
normal approximation to estimate this
00:04:20
probability we could calculate the exact
00:04:22
probability from the binomial
00:04:23
distribution using a computer or even by
00:04:25
hand if we had to but we're going to use
00:04:27
the normal approximation here and we're
00:04:30
going to say that the probability that X
00:04:31
is bigger than or equal to 52 this is
00:04:33
going to be approximately the
00:04:35
probability that Zed is bigger than or
00:04:38
equal to we could say 52 minus the mean
00:04:42
45 we're simply standardizing like
00:04:44
normal divided by the standard deviation
00:04:47
Square < TK of 18 and this is equal to
00:04:49
the probability that Zed takes on a
00:04:53
value that's at least as big as 1.645
00:04:56
rounded to three decimal places then we
00:04:59
would go to a computer or our standard
00:05:01
normal table here's
00:05:03
1.645 and we're interested in this area
00:05:06
so I'm going to leave that up to you to
00:05:08
verify for yourselves that that's going
00:05:10
to be approximately
00:05:13
0.495 if you use a standard normal table
00:05:16
instead of a computer you might get a
00:05:17
rounded version of that but you should
00:05:19
get pretty close to that value now
00:05:21
compare that with the exact value that
00:05:24
we get if we use the binomial formula
00:05:26
which I'm not showing here but I used a
00:05:28
computer to get the exact value based on
00:05:30
the binomial distribution and we get
00:05:33
0611 and our value based on the normal
00:05:36
approximation well it's in the ball park
00:05:38
but it's still a little bit
00:05:40
off but we can improve that
00:05:43
approximation with something we call a
00:05:46
continuity correction we are moving from
00:05:48
this discrete binomial distribution to
00:05:51
this continuous normal distribution and
00:05:54
when we do that we can improve the
00:05:56
approximation with this continuity
00:05:59
correction
00:06:01
to illustrate let's look what's going on
00:06:02
here this is a plot of the binomial
00:06:04
distribution with n is 75 and P is 6 and
00:06:08
the Shaded green part is the probability
00:06:11
that we're interested in the probability
00:06:13
that we get a value of 52 or
00:06:17
greater what if I superimpose the normal
00:06:20
curve here's the superimposed normal
00:06:23
curve and this red shaded area is the
00:06:27
probability calculation that we carried
00:06:29
out on the last page and here is the
00:06:31
value 52 now in the binomial setting 52
00:06:36
means something 52 successes out of 75
00:06:40
but on the continuous front for a
00:06:43
continuous random variable 52 means
00:06:46
52.00
00:06:48
0000000000 infinite Zer there and it is
00:06:52
distinctly different from
00:06:55
52.1
00:06:56
say now I've blown this part up on the
00:06:59
next slide just to make it a little
00:07:00
easier to
00:07:01
see so to come closer to regaining our
00:07:04
original meaning of 52 we're going to
00:07:06
say okay 52 in the discret sense had
00:07:08
some meaning 52 in the continuous sense
00:07:11
means exactly 52
00:07:13
52.00 so to regain that original meaning
00:07:16
we should let 52 Take on all values
00:07:19
between 52 and 1/2 and 51 and 1/2 that
00:07:23
way it comes closer to representing what
00:07:25
is intended here in the binomial
00:07:27
distribution and so as we can see here
00:07:29
here right at 52 when we started at
00:07:32
52.00 and we did our probability
00:07:35
calculation we were really missing out
00:07:37
an important part we were missing out
00:07:40
this half of 52 in a sense and so what
00:07:42
we really should do is start here at
00:07:47
51.5 that's where we should start so
00:07:50
that's what it looks like on the next
00:07:51
page if I start at 51.5 my approximation
00:07:56
my red shaded area here is going to be a
00:07:58
lot closer to the total those green
00:08:00
probabilities and a lot closer to
00:08:02
reality so if I want my probability that
00:08:05
X is bigger than or equal to 52 before
00:08:08
doing the normal approximation I am
00:08:11
going to use the continuity correction
00:08:13
and say this is the probability that Zed
00:08:16
is bigger than or equal to
00:08:19
51.5 minus the mean over the standard
00:08:25
deviation this works out to the
00:08:27
probability that Zed is bigger than than
00:08:30
or equal to 1.53
00:08:33
2 and we go to our standard normal table
00:08:36
or a computer or what have you and if we
00:08:39
did this without any roundoff error and
00:08:42
just rounded our final answer to four
00:08:44
decimal places we'd see that this is
00:08:47
0.628 and as I'll illustrate in a little
00:08:50
bit that's closer to the true
00:08:52
probability based on the binomial
00:08:54
distribution than when we didn't use the
00:08:56
continuity
00:08:58
correction in this new question we're
00:09:00
interested in the probability that X is
00:09:02
strictly greater than 52 that is what
00:09:05
the Shaded green bits represent and to
00:09:08
be strictly greater than 52 I want to
00:09:11
make sure I don't include any of 52 so I
00:09:14
really should start right here I should
00:09:16
start at
00:09:18
52.5 let's see what that shaded area
00:09:21
under the normal curve looks
00:09:23
like that looks a little bit better and
00:09:25
so I want my probability that X is
00:09:27
strictly greater than 52 this is going
00:09:30
to be approximately the probability that
00:09:32
Zed is greater than
00:09:37
52.5 minus my mu which is 45 divided by
00:09:41
the standard deviation which is the
00:09:43
square < TK of 18 this works out to the
00:09:45
probability that Zed is greater than
00:09:49
1768 and if we found that value under
00:09:52
our standard normal curve 1. 768 and we
00:09:55
did it without any roundoff error and
00:09:57
rounded our final answer to for places
00:10:00
we would see that's the answer now you
00:10:02
should be able to come close to that but
00:10:04
not exactly that from a standard normal
00:10:08
table now what about here let's say I
00:10:10
wanted the probability that X is less
00:10:13
than or equal to 52 that's what that
00:10:15
shaded green bit
00:10:16
represents and I can't start exactly at
00:10:19
52 that wouldn't be quite right to
00:10:20
include all of 52 and go left I need to
00:10:23
start here I need to start at 52.5 so
00:10:27
let's see what that looks like when
00:10:28
shaded under the normal
00:10:30
curve that looks like it might provide
00:10:32
us a pretty reasonable approximation so
00:10:33
when I want the probability that X is
00:10:35
less than or equal to
00:10:37
52 that's going to be approximately with
00:10:40
my continuity correction I want to
00:10:42
include all of 52 and so I should start
00:10:45
at
00:10:47
52.5 subtract the mean which is 45 and
00:10:50
divide by the standard
00:10:52
deviation this is going to be equal to
00:10:54
the probability that Zed is less than or
00:10:57
equal to 1.7
00:11:00
68 and if we plot that
00:11:03
out
00:11:06
1768 that's this entire area
00:11:10
here and that works out to
00:11:13
0.961
00:11:15
5 what about if we're interested in the
00:11:18
probability that X is strictly less than
00:11:21
52 which is represented by the green
00:11:23
shaded bit if I'm going strictly less
00:11:27
and I don't want to include 52 then I
00:11:29
should start here at
00:11:31
51.5 let's see what that looks like when
00:11:33
shaded in that looks like that might
00:11:36
give us a pretty darn reasonable
00:11:37
approximation so if I want my
00:11:39
probability that X is less than 52 then
00:11:42
I'm going to say that's approximately
00:11:43
the probability that Zed is less than
00:11:47
51.5 minus mu over
00:11:52
Sigma and this is equal to the
00:11:54
probability that Zed is less than 1.53
00:11:59
two and if we put that into our
00:12:02
computer and we did this without any
00:12:04
roundoff error and then just rounded to
00:12:07
four decimal places at The Bitter End we
00:12:09
would say that this is 0.
00:12:16
9372 so let's look at a summary of what
00:12:19
we just did if I want the probability
00:12:21
that X is bigger than or equal to 52
00:12:24
then I want to include all of 52 and go
00:12:27
right which means I should start at
00:12:31
51.5 if I want the probability that X is
00:12:34
strictly greater than 52 well I'm going
00:12:36
right but I don't want to include any of
00:12:39
52 so I should start at
00:12:42
52.5 if I am going left less than or
00:12:46
equal to 52 I want to include all of 52
00:12:50
while going left which means that I
00:12:52
should start at
00:12:54
52.5 and if I'm going strictly less than
00:12:57
52 I want to make sure I don't include
00:12:59
any of 52 while going left that which
00:13:02
means I should start at
00:13:04
51.5 now I strongly recommend that you
00:13:06
don't simply memorize this but you try
00:13:08
to follow the underlying logic if you
00:13:10
truly understand why we're doing what
00:13:12
we're doing here then it'll be easier to
00:13:14
do properly when you have to now let's
00:13:17
take a quick look here at how that
00:13:19
continuity correction improved the
00:13:21
approximation for a couple of cases that
00:13:23
we just looked at if we're interested in
00:13:25
the probability that X is bigger than or
00:13:27
equal to 52 without the continuity
00:13:29
correction we got
00:13:31
0495 but with the continuity correction
00:13:34
we got
00:13:35
0628 which is much closer to the exact
00:13:39
value based on the binomial distribution
00:13:41
that's what this exact value is coming
00:13:43
from similarly down here when we wanted
00:13:46
greater than 52 we had
00:13:48
0495 we use the same calculation in both
00:13:51
cases we didn't use our continuity
00:13:52
correction and with the continuity
00:13:55
correction we get
00:13:57
0385 which is much closer to the exact
00:14:00
value based on the binomial distribution
00:14:02
so the continuity correction has greatly
00:14:05
improved our approximation