So last time, we discussed how to find the
probabilities underneath the normal curve for
three cases. If the point lies in the lower tail,
as this one, or upper tail, or between. So we can
do the computations for this kind of
probabilities. And I think we gave two examples.
For example, if we are looking for probability of
Z is smaller than 0.1, and Z, as we mentioned last
time, is the standardized normal distribution that
has mean of 0 and sigma is 1. In this case, just
go to the table you have. Now we are looking for 0
.12, for example. So here we have 0.1. Under 2, we
get this result. This value is the probability of
Z smaller than 0.5. But you have to keep in mind
that we have to transform first from normal
distribution to standardized normal distribution.
For this specific example, if we are looking for
mean of 8 and standard deviation of 5, that's for
the normal distribution. In this case, the z-score
is given by this equation, which is x minus 3
divided by sigma. So 8.6 minus 8 divided by 5
gives 0.12. In this case, we can use the
standardized normal table. Now, for the other
case, we are looking for the probability of x
greater than 8.6. So we are looking for the upper
tail probability. The table we have gives the area
to the left side. And we know that the total area
underneath the normal curve is 1. So the
probability of x greater than 8.6 is the same as 1
minus the probability of x smaller than x less
than 8.6. So here, for 8.6, we got z squared to be
0.12. So the probability of z greater than 0.12 is
the same as 1 minus z of z is smaller than 0.12.
So 1 minus the answer we just got from part A will
get us to something. So this is the probability of
X is greater than 8.6. So all of the time, if you
are asking about computing the probability in the
upper tail, you have to first find the probability
in the lower tail, then subtract that value from
1. So that's the probability for the upper tail.
The last one, we are looking for probability
between two values. For example, x. What's the
probability of x greater than 8 and smaller than 8
.6? So we are looking for this area, the red area.
So the probability of x between these two values
actually equals the probability of x smaller than.
8.6 minus the probability of X smaller than 8.
And that, in this case, you have to compute two
values of this score. One for the first value,
which is A. This value is zero because the mean is
zero. And we know that Z can be negative. If X is
smaller than Mu, Z can be positive if X is greater
than Mu and equals zero only if X equals Mu. In
this case, X equals Mu, so Z score is zero. The
other one as we got before is 0.12. So now, we
transform actually the probability of X between
8 and 8.6 to z-score between 0 and 0.12. In this
case, we can use the normal theorem. Now, this
area is, as we mentioned, just b of x smaller than
0.12 minus b of x, b of z smaller than 0. This
probability, we know that, 0.54878. Now, the
probability of z smaller than 0 is one-half.
Because the total area underneath the normal curve
is 1, and 0 divided the curve into two equally
parts. So the area to the right of 0 is the same
as the area to the left of 0. So in this case,
minus 0.5. So this is your answer. So the
probability of Z between 8 and 8.6 is around 0.0478.
I think we stopped last time at this point.
This is another example to compute the probability
of X greater than 7.4 and 8. Also, it's the same
idea here, just find these scores for the two
values.
L with B of Z greater than minus 0.12 up to 0. Now
this red area equals the area below 0. I mean B of
Z smaller than 0 minus the probability of Z
smaller than minus 0.12. Now by using symmetric
probability of the normal distribution, we know
that the probability of Z smaller than minus 0.12
equals the probability of Z greater than 0.12.
Because this area, if we have Z smaller than minus
0.12, the area to the left, that equals the area
to the right of the same point, because of
symmetry. And finally, you will end with this
result.
D of z minus 0.12, all the way up to 0, this area,
is the same as the area from 0 up to 0.5. So this
area actually is the same as D of z between 0 and
0.5. So if you have a negative sign, and then take
the opposite one, the answer will be the same
because the normal distribution is symmetric
around 0. The questions, I think we stopped
here. And also we talked about empirical rules.
The one we mentioned in chapter three, in chapter
three. And we know that, as we mentioned before,
that is 68.16% of the observations fall within one
standard deviation around the mean. So this area
from mu minus one sigma up to mu plus sigma, this
area covers around 68%. Also
95% or actually 95.44% of the data falls within
two standard deviations of the mean. And finally,
around most of the data, around 99.73% of the data
falls within three subdivisions of the population
mean.
And now the new topic is how can we find the X
value if the probability is given. It's vice
versa. In the previous questions, we were asking
about find the probability, for example, if X is
smaller than a certain number. Now suppose this
probability is given, and we are looking to find
this value. I mean, for example, suppose in the
previous examples here,
Suppose we know this probability. So the
probability is given. The question is, how can we
find this value? It's the opposite, sometimes
called backward normal calculations.
There are actually two steps to find the x value
for a certain probability or for a given or for a
known probability the first step we have to find
the z score then use this equation to find the x
value corresponding to the z score you have and x
is just mu plus sigma times mu so first step you
have to find the z score
corresponding to the probability we have. So find
the z value for the non-probability, then use that
z score to find the value of x by using this
equation. So x equals mu plus z sigma. z could be
negative, could be positive, depends on the
probability you have. If the probability is above
0.5, I mean 0.5 and greater than 0.5, this
corresponds to z positive. But if z-score is negative, I'm
sorry, if z-score is negative, then the
probability should be smaller than 0.5. So if the
probability is given less than 0.5, then your z
-score should be negative, otherwise should be
positive. So you have to be careful in this case.
Now look at this example. Let x represent the time
it takes in seconds to download an image file
from the internet. The same example as we did
before. And here we assume that x is normal
distribution with mean of 8 and standard deviation
of 5. Now, let's see how can we find the value of
x such that 20% of download times are smaller than
x.
So, this probability is a fraction. Also, always
the probability is between 0 and 1. So, the
probability here is 20%. In this case, your z
-score should be negative. Because 20% is more
than 0.5. So, z-score should be in this side, in
the left side.
So, again, he asks about finding x-value such that
20%.
So here again we are looking for this value, for
the value of x, which is smaller than the area to
the left of this x, equals 0.2.
Now, the first step, we have to find the z-score.
It's backward, z-score first, then x. Find a z
-score corresponding to the probability of 0.2.
The approximate one, the near value, I mean, to
the 0.2 is 0.2005. Sometimes you have the exact
value from the table you have, but most of the
time you don't have it. So you have to look at the
approximate value, which is very close to the one
you have. So here, we are looking for 0.2. The
closest value to 0.2 is 0.2005. Now, the
corresponding value to this probability is minus 0
.8 all the way up to 4. So your z-score is
negative 0.84. So this is the first step. Any
question? Again.
Now if we just go back to this equation,
z equals x minus mu over sigma. A cross
multiplication, I mean if you multiply both sides
by sigma, you will get sigma times z equals x
minus mu.
Now, in this question,
he asks about, find the value of x such that 20%
of download times are less than x.
Now the probability is less than 0.5, so your z
-score should be on the left side. So here we need
to find the value of z first. Go back to the
normal table you have.
This is the normal table.
We are looking for minus 0.2. I'm sorry, we are
looking for 0.2. So the closest value to 0.2 is
this one, 0.2005. So this is the closest value.
So the exact answer is sometimes not given. So the
approximate one, minus 0.8, all the way up to 4.
So z-score minus 0.8. Any question?
So the value of z-score is minus 0.84. So my 190 00:15:
So z-score is 1. So we are looking for the
probability of z greater than 1.
1.5. 1.2. 1.5. 1.5. 1.5. So I'm looking for the
probability of x of z greater than
1 minus P
of Z less than or equal to 1.5. Now go back to the
table.
Now 1.5 under 0. It's 0.9332.
So, 1 minus this probability gives 0.0668.
That's the probability of X greater than 4.4.
So, the answer is 0.0668.
Now, for the same question.
What's the probability that a randomly selected
fish will weigh between 3 and 5 pounds?
3?
Up to 5. So first we have to find the z-score for 3
out of 5. For it to be just 3 minus 3.2. Divide by
0.8 is the first z value. Negative 0.2 divided by
0.8 minus 0.25. The other one, 5 minus 3.2 divided
by 0.8. 1 minus 0.8 divided by 0.8 equals
2.25.
Just double check this result. So here, the
probability of X between 3 and 5 equals minus 0
.25, smaller than Z, smaller than 2.5.
So it's transformed from normal distribution to
standardized normal distribution. So here instead
of computing the probability of X between three
and five, we are looking for the probability
between Z between actually minus. It's minus
because your value here is smaller than the
average. The average was 3.2, so it should be
negative. So z score between minus 0.25 all the
way up to 2.25. So now, this is the probability we
are looking for. Zero in the middle minus one
-fourth to the left of z of zero, mu of zero. And
this is the value of 2.25. Now we are looking
actually for this probability.
The area between minus 0.25 all the way up to 2.5.
So this area equals the
probability of Z less than 2.25 minus.
And again, use the normal. table to give this
value and another one.
Any questions? So first step here, we compute the
z-score for each value x. So the problem is
transformed from normal distribution to
standardized normal distribution. So it becomes z
between minus 1.25 up to 2.25. Now, this area,
this dashed area equals the area below 2.25 minus
the area below minus 1.25. Now, by using the
similar way we did before, you will compute the
value of z. The probability of z is smaller than 2
.25 by using The normal table. So here, 2.2 up to
5. So 9, 8, 7, 8. 9, 8, 7, 8.
So the area below 2.25, 2.2, this value. All the
way up to 5 gives 987.
Now, what's about the probability of Z smaller
than minus 0.25? If you go back to the Z table,
but for the other one, the negative one.
Minus 2 minus 0.2 up
to 5. 0.4013 minus,
that will give the probability between three and
five.
This is the second part.
So the final answer.
So this is the probability that the selected fish
will weigh between three and five pounds.
Now, other question is, for the same problem, you
said a citation Catfish should be one of the top 2
% in the weight. Assuming the weights of catfish
are normally distributed, at what weight in bounds
should the citation, the notation be established?
So in this board, he asked about what's the value
of x, for example.
is greater than what value here. And this
probability equals 2%. Because you said the
citation catfish should be one of the top 2%. So
the area in the right here, this area is 2%.
What's the value of x in this case? So here, the
value of x greater than a equals 0.02, and we are
looking for this value.
gives the area to the left side. So this
probability actually, the area to the right is 2%,
so the area to the left is 98%. So this is the
same as,
as we know, the equal sign does not matter because
we have continuous distribution.
continuous distribution, so equal sign does not
matter. So now, if you ask about P of X greater
than a certain value equals a probability of, for
example, 0.02, you have to find the probability to
the left, which is 0.98, because our table gives
the area to the left. Now, we have to find the
value of A such that Probability of X is more than
or equal to 0.98. So again, we have to look at the
normal table, but backwards, because this value is
given. If the probability is given, we have to
look inside the body of the table in order to find
the z-score.
x equals mu plus z sigma in order to find the
corresponding value x. So again, go back to the
normal table, and we are looking for 98%. The
closest value to 98%, look here, if you stop here
at 2, go to the right,
Here we have 9798 or
9803.
So the answer might be your z-score could be 2.05
or 2.06. So again, in this case, the table does
not give the exact. So the approximate one might
be between them exactly. Or just take one of
these. So either you can take 9798, which is
closer to 98% than 9803, because it's three
distant apart. So maybe we can take this value.
Again, if you take the other one, you will be OK.
So you take either 2.05. or 2.06. So let's take
the first value, for example. So my x equals mu, z
is 2.05, times sigma, 0.8. Multiply 2.05 by 8, 0
.8, then add 3.2, you will get What's your answer?
3.2 plus 2 point... So around 4.8. So your answer
is 4.84.
Now if you go back to the problem, and suppose you
know the value of x.
So the probability of X.
Double check to the answer.
4.84. Just check. V of X greater than this value
should
be Two percent. Two percent. So the probability of
X greater than this value should be equal to one
zero. So this problem is called backward normal
calculation because here first step we find the
value of this score corresponding to this
probability. Be careful. The probability of X
greater than 2 is 0.02. So my value here should be
to the right. Because it says greater than A is
just 2%. If you switch the position of A, for
example, if A is on this side, And he asked about
E of X greater than E is 2%. This area is not 2%.
From A up to infinity here, this area is not 2%
because at least it's greater than 0.5. Make
sense? So your A should be to the right side.
Because the value of X greater than E, 2% is on
the other side. Let's move to the next one.
For the same question.
Again, the owner of Catfish Market determined the
average weight of a catfish 3.2 with
standardization 0.8 and we are assuming the
weights of catfish are normally distributed, kiosk
above. Above what weight? Do 89.8% of the weights
care?
Above? And above, so x greater than. X minus. And
98, 89, sorry, 89. So this is a percentage he's
looking for. 89.8%. Now what's the value of A? Or
above what weight? Do 89.8% of the weights occur?
You just make the normal curve in order to
understand the probability. Now, A should be to
the right or to the left side? Imagine A in the
right side here. Do you think the area above A is
89%? It's smaller than 0.5 for sure. So it should
be the other side. So this is your 8. Now, this
area makes sense that it's above 0.5. It's 0.8980.
Now, B of X greater than equals this value. And
again, the table gives the area to the left. So
this is actually X less than A, 1 minus this
value, equals 0.1020.
Now go back to
the normal table. Here it's 0.1020. So it should
be negative. I mean, your z-scope should be
negative.
Now look at 0.102.
Exactly this value. 0.102 is minus 1.2 up to 7. So
minus 1.27.
Minus 1.2. All the way up to 7 is 0.102. So the
corresponding z-score is minus 1.17. Now x again
equals mu plus z sigma.
Mu is 3.2 plus z is negative 1.17 times sigma.
So it's equal to 3.2 minus 127 times 0.3. By
calculator, you'll get the final result.
If the probability is smaller than 0.5, then this
one is negative. Go to the other one. If the
probability is above 0.5, then use the positive z
-score. So what's the answer? 2.18.
Be careful. In the previous one, We had a
probability of X greater than A equal 2%. In
this case, the value of A, for example, is located
in the upper tail here.
For this part, you ask about B of X greater than A
equal 0.89. It's here more than 0.5
Below minus 3, I mean smaller than minus 3, or
above 3, these points are suspected to be
outliers.
So any point, any data value smaller than minus 3
in this form, or above plus 3 is considered to be
an outlier. If we compute the lower limit, which
is Q1 minus 1.5 IQR over the upper limit.
So we
have lower limit, upper limit. So lower limit.
And upper limit.
Any data point below lower limit or above upper
limit is considered to be a type. So therefore, we
have two methods to determine or to examine if the
observation is enough there or not. One by using
this score, straightforward. And the other one, we
have to look for the lower limit and upper limit.
The other method by using software and later on
you will take SPSS in order to determine if the
data is normally distributed by using something
called QQ plot or normal probability plot. So I'm
going to skip this part. Because data is taken by
using software. But in general,
you may look at this graph. Generally speaking, if
you have a probability plot of a data, and the
points lie on a straight line, or close to it, in
this case, the distribution is normal. It's hard
to make this graph manual. It's better to use
software. But at least if we have this graph, and
the points are close to the straight line. I mean,
the points are either on the straight line, lies
on the straight line, or close to it. In this case,
the data is normally distributed. If the data
points scattered away of the straight line, then
the distribution is not normal, either skewed to
the right or skewed to the left. So for this
specific graph, the plot is normally distributed,
approximately normally distributed. Because most
of the points here lie close to the line and few
are scattered away. Or it means that there are few
outliers in this case, we can ignore these values.
So here the plot is approximately a straight line
except for a few outliers at the low and the
right, those points. So generally speaking, the
distribution is normal distribution. That's all
for this chapter.