00:00Hi guys, so in the
00:01last video, the last lesson
00:03what we did is we
00:04had two variables and we
00:09scaled one of the variables
00:11to give us a nice
00:13smaller numbers and we ended
00:15up showing that we had
00:17a linear relationship between one
00:19of the variables, so let's
00:20show you here. So between
00:21this variable, the number of
00:24years and this variable scaled,
00:26so between x and ln
00:27of y
00:28why we had a linear
00:29relationship. And we showed that
00:33if there's a linear relationship
00:35between x and ln of
00:39y, then the relationship between
00:41x and y is an
00:43exponential relationship. And we sometimes
00:49call that a semi -log
00:51graph, the graph between this
00:54variable here and the log
00:56this variable, in this case
00:57the natural log with this
00:58variable, it's a semi log
01:00graph if we have a
01:01this linear relationship. Well, it's
01:04a semi log graph anyway,
01:05but if it's that if
01:06we have the linear relationship
01:07between this and this, we
01:09have an exponential relationship between
01:11this and this. Now, same
01:14kind of thing here, this
01:15time it's log log graph,
01:17we are going to get
01:18the log of this variable
01:19and the log of this
01:21variable. And we're going to
01:22show that it is a
01:23linear relationship.
01:24And if we get a
01:25linear relationship between the log
01:26of this and the log
01:27of this, then we will
01:31see what kind of relationship
01:33we get between t and
01:34d. I'm not going to
01:35show it because we're going
01:36to do it in the
01:37last. I'm not going to
01:38tell you now because we're
01:39going to do it as
01:40the last part of this
01:41question. OK. So firstly, let's
01:46get the log of both.
01:47Well, let's look at the
01:48example. The example is a
01:49basketball is dropped from the
01:50top of a building
01:52This is from the top
01:53of the building is given
01:54below after seconds 1 through
01:565. So this basically means
01:581 second, it's 3 .5
02:00meters, 2 seconds 15 .6,
02:035 seconds 114 .5. Clearly,
02:06it isn't a linear relationship
02:07between T and D. If
02:09you think about dropping a
02:10basketball, it wouldn't be anyway
02:12because of gravity. There is
02:16the basketball will accelerate. Now,
02:18if you dropped it from
02:19a tall enough building,
02:20it might approach, yeah, it
02:22might become a linear relationship,
02:23but that's not what's happening
02:24here. Okay, so first thing
02:26we're gonna do is get
02:28ln of t and ln
02:29of t because that's what
02:30they ask you to do.
02:31Remember, it doesn't have to
02:32be the natural log, but
02:34that's what they're asking us
02:35to do in this question.
02:36So I've cheated again guys,
02:38I've put in the, I've
02:40put in t and I've
02:41put in d. I'm gonna
02:42do equals ln of this,
02:47per center.
02:48and drag it down. I
02:50did this in the last
02:51lesson. And then here I'm
02:53going to say equals ln
02:55of this, press enter, drag
03:02it down. That's not what
03:05I wanted. That's definitely not
03:07what I wanted. Okay, there.
03:09Let's call this, this is
03:12ln t. And let's call
03:14this ln
03:16of D, presenter. Okay, I'm
03:20going to totally cheat here
03:21and do this. I've already
03:22written in the numbers. Okay,
03:26so it says the relationship
03:27between L and F T
03:28and L and F D
03:30can be modeled using the
03:31relationship with this. So there's,
03:34we're going to see that
03:34there's a linear relationship between
03:37this and this. And let's
03:38actually show it using the
03:41calculator here. So if I
03:43go to doc
03:44event. Sorry, menu, no document.
03:52I want, I want to
03:53do insert, sorry, insert. I
03:58want data and statistics. So
04:01this is my data here.
04:04Let's go with, let's do
04:06d, let's do t here,
04:09and then d here, see
04:11what's happening.
04:12Okay, so that's I'm sure
04:15it looks a bit linear,
04:17but maybe my e might
04:20think it's exponential. I don't
04:22know let us change this
04:26now to ln of t
04:28and change this one to
04:32ln of t now that
04:35I Don't know I think
04:36that's a little bit more
04:38of an exponential Sorry
04:40a linear relationship. So let's
04:43go back to this. Let
04:46us get menu statistics, stack
04:51calculations, linear regression, mx plus
04:55b. So the x list
04:58is now my ln of
05:01t. And my wireless is
05:03my ln of d. That
05:06can all stay the same.
05:07That can go here.
05:08Press okay. So my M
05:10is 2 .19907. So M,
05:15that's right, this here. So
05:17for part A, M, M
05:21equals 2 .19907, 2 .19907.
05:28And B equals 1 .27622,
05:341 .27622,
05:362, power b. Find the
05:39correlation coefficients. That just means
05:40between these two. The correlation
05:42coefficient is here. It's, or
05:450 .997, 224, 0 .997,
05:52224. Finally, using your equation,
05:56find an expression for d
05:58in terms of t. OK,
05:59let's do this. So c.
06:02I now have ln of
06:04d.
06:04is equal to, so I
06:06have ln of d is
06:09equal to m, that's 2
06:11.1997, ln of t plus
06:17b, so plus 1 .27622,
06:22but I want just d
06:23in terms of this, so
06:26I can say d is
06:27equal to e to the
06:29power of all of this,
06:31e to the power of
06:322 .1997,
06:33907, Ln of t plus
06:381 .276, 2, 2. OK,
06:45let's put a space here.
06:47So this d is equal
06:49to this is e to
06:53the power of 2 .1997,
06:56Ln of t. That's the
06:57same as e to the
06:58power of Ln of t.
07:01to the power of 2
07:02.1997. So this is my
07:09power rule of logs. That's
07:11like if I have 3l
07:14and a 4, I can
07:15say that's ln of 4
07:18cubed. I'm allowed to, I'm
07:19allowed to bring the 3
07:20up here. So similarly, what
07:23I've done here is I've
07:24brought the 2 .19907 up
07:26here. And then it's
07:29plus, so it's e to
07:30the power of this times
07:33e to the power of
07:35this 1 .27622. So, e
07:40to the power of this
07:41plus this is e to
07:42the power of this times
07:43e to the power of
07:44this. This is my e
07:44to the power of a,
07:46or e to the power
07:47of a plus b is
07:48equal to e to the
07:49power of a, e to
07:51the power of a times
07:53e to the power of
07:55b.
07:57Okay, nearly done. What is
07:59e to the power of
08:001, 2, 1, 2, 7,
08:036, 2, 2? Let's find
08:04that. Let's find that out
08:05right now. So I have
08:10e to the power of
08:131 .27622, 3 .58307, 3
08:21.58307,
08:25And then it's e to
08:28the power of ln of
08:29this. But you know e
08:33to the power of ln
08:34of x is equal to
08:36x. So e to the
08:37power of ln of t
08:39to the power of 2
08:40.9, 2 .19 or 7
08:44is just going to be
08:46t to the power of
08:492 .19, 907 is just
08:51going to be t to
08:51the power of 2 .19,
08:51907.
08:53That is the relationship between
08:57D and T. It is
08:583 .58307 times T to
09:03the power of 2 .1997.
09:11And that's it. So when
09:14you have a linear relationship
09:17between the two
09:21scale variables. So like the
09:23log of one variable and
09:25the log of another variable
09:26when there's a linear relationship
09:27between the two and there
09:28is because the correlation coefficient
09:30is 0 .9 and 7,
09:312, 2, 4. Then you
09:34end up with a power
09:35relationship between the two original
09:38variables. So here I have
09:39this is the power relationship.
09:42D is equal to this
09:44times t to the power
09:46of 2 .19907.
09:49Okay, that's it done. That's
09:53the lesson guys. Hopefully that
09:55makes sense. Not that straightforward,
09:59but I think once you
10:00get your head around a
10:03log log and semi -log
10:05graphs and what they mean
10:07and what it means to
10:10find a linear relationship between
10:13those and how that converts
10:16to a different type of
10:17relationship between the original variables.
10:20Hopefully it makes sense. Okay
10:21guys, that's it. I will
10:22see you in the next
10:24video.