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Current time:0:00Total duration:7:40

AP.STATS:

DAT‑1 (EU)

, DAT‑1.J (LO)

, DAT‑1.J.1 (EK)

- [Lecturer] We are told
that a conservation group with a long-term goal
of preserving species believes that all at-risk
species will disappear when land, inhabited by
those species, is developed. It has an opportunity to
purchase land in an area about to be developed. The group has a choice of
creating one large nature preserve with an area of 45 square kilometers and containing 70 at-risk species, or five small nature preserves, each with an area of
three square kilometers and each containing 16 at-risk species unique to that preserve. Which choice would you recommend and why? There are some interesting data here. It looks like some data they have gathered for different islands. We have their areas. This is the number of
species at risk in 1990. The species extinct by 2000. We can see for these various islands, we can see their areas and the proportion that got extinct. It looks like they're
plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. The vertical axis is the
proportion extinct in 2000. It's these numbers. But the horizontal axes isn't
just a straight-up area. It's the natural log of the area. Why did they do this? Notice, when you make the horizontal axis the natural log of the area, it looks like there is
a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our
tools of linear regression to think about what
would be the proportion extinct for the 45 square kilometers versus for the five small
three-kilometer islands. Pause this video and see if you can figure
it out on your own. They gave us the regression data for a line that fits this data. All right. Now let's work through it together. To make some space because
all of it is already plotted right over here, and we
have our regression data. The regression line, we know
it's a slope in y-intercept. The y-intercept is right
over here, 0.28996. 0.2, this is, let's see,
one, two, three, four, five. 28996. It's almost 29. It's gonna be right over here would be the y-intercept. Its slope is negative 0.05 approximately. I could eyeball. It probably is gonna
look something like this. That's the regression line. Or another way to think about it is the regression line tell
us in general the proportion, proportion, obviously a proportion, shorthand for proportion extinct, is going to be equal to
our y-intercept 0.28996 minus 0.05323. We have to be careful here. You might be tempted
to say times the area, but no, the horizontal axis here is the natural log of the area. Times the natural log of the area. We can use this equation
for both scenarios to think about what is
going to be the proportion that we would expect to get
extinct in either situation, and then how many actual
species will get extinct. The one that maybe has fewer
species that get extinct might maybe the best one, or the one that the more
that we can preserve is maybe the best one. Let's look at the two scenarios. The first scenario is the
45 square kilometer island. This is just one, so times one. What is gonna be the proportion, proportion that we would
expect to go extinct? Based on this regression, it's going to be 0.28996 minus 0.05323 times the natural log of 45. If we want to know the actual
number that go extinct, so number extinct would be
equal to the proportion, would be equal to the proportion times how many, let's see,
the 45 square kilometers and it contains 70 at-risk species, so times our 70 species. We can get our calculator
out to figure that out. This is the proportion we
would expect to go extinct in the 45 square kilometer island based on our linear regression. This would be equal to. It looks like almost 9%. If we want to figure out the actual number we would expect to go extinct, we would just multiply that
times the number of species on that island, so times 70, and we get approximately about 6.11. Let me write that down. This is going to be approximately 6.11. We could say there would be approximately if we, let's just say six extinct, and this is all very approximate. Extinct. Approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves. So it's going to be
three square kilometers times five islands. We're gonna just do the same exercise. Our proportion that goes
extinct is gonna be 0.28996, that's just the y-intercept
for our regression line, minus 0.05323, and you
have a negative sign there 'cause we have a negative slope, and this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers. Three square kilometers. Our number extinct, Our number extinct is going to be equal to our proportion that we will calculate
in the line above times. Let's see. Five small nature preserves, each with an area of
three square kilometers and each containing 16 at-risk species. Five times 16, if each island has 16 and there's five islands,
that's going to be, five times 16 is 80. Times 80. Let's figure out what this is. Get the calculator out again. We are going to get. This is going to be the proportion. It's a much higher proportion. We'll multiply that times
our number of species, so times 80 to figure out how
many species will go extinct. We have here it's approximately 18.52. This is approximately 18.52. Another way to think about it is we're gonna have approximately. If we round let's just say
19 extinct, 19 extinct. And then if we have 19 extinct, how many are we gonna save? We're gonna have 61 saved. 61 saved. Even if you said 18
1/2 here and 61.5 here, on either measure, the 45
square, the big island is better. You're gonna have fewer
species that are extinct and more that are saved. Which choice would you recommend and why? I'd recommend the one large island because you're gonna save, you would expect to save more species, and you would expect that
fewer are going to get extinct based on this linear regression.

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